Can general relativity be completely described as a field in a flat space? Can general relativity be completely described as a field in a flat space? Can it be done already now or requires advances in quantum gravity?
 A: From Box 17.2 of MTWs "Gravitation", section 5:

Einstein's geometrodynamics viewed as the standard field theory for a field of spin 2 in an "unobservable flat spacetime" background.
...
f. "This series continues indefinitely, and sums to the full non-linear Einstein equations,
...
i. Deser summarizes the analysis at the end thus:  "Consistency has
  therefore led us to the universal coupling, which implies the
  equivalence principle.  It is at this point that the geometric
  interpretation of general relativity arises, since all matter now
  moves in an effective Riemann space...
[The] initial flat 'background' space is no longer observable.

The original work on the topic was Deser 1970. An update is given in Deser 2010.
Deser, Gen Rel Grav 1 9(1970), http://arxiv.org/abs/gr-qc/0411023
Deser, Gen.Rel.Grav.42:641-646,2010, http://arxiv.org/abs/0910.2975
A: A more recent alternative to Deser's work is that of Gull, Doran, and Lasenby. Framed in the mathematical of geometric (real Clifford) algebra and its associated calculus, this framework presents gravity as a gauge field on a Minkowski spacetime.  The formulation is clearly inspired by relativistic quantum mechanics and tetrad approaches, but it has some unique characteristics.  In particular, there are some fine differences from GR in some theoretical consequences: for example, it's not possible to have all four quadrants of the Kruskal extension in GTG, and the authors remark that this is a common feature distinguishing a second-order theory (GR) from a first-order theory (GTG).
In large part, however, GTG greatly agrees with GR, and it is derived largely from common arguments about invariance with respect to diffeomorphisms and local Lorentz rotations. The authors believe this leads to a cleaner interpretation of the equivalence principle and general covariance, as changes of coordinates can be neatly separated from physical changes in the gauge field.  For my part, these arguments seem particularly transparent compared to any idea of infinite series that converge to GR.
A: ****(Added a long clarification at the end about the work of Deser and others as mentioned in other answers which may seem to contradict this one)**.
ADDITION: in the paper by Straumann, N. - Reflections on Gravity (ESA-CERN Workshop on Fundamental Physics in Space and Related Topics, European Space Agency, 5-7 April 2000(2001), SP-469,55), it is shown (similarly to Deser's classical work but in a more expository style) that a spin-2 field theory on a flat Minkowki 4-manifold ends up being totally equivalent to Einstein's curved spacetime where the ab inition Minkowski metric ends up being unobservable whereas the observable and physical metric is dynamical supporting the relational meaning advocated in the rest of this answer which emphasizes the idea that space, time and causal structure are relational notions between dynamical entities and NOT an absolute static stage (be it Newton's or Minwkoski's) where things, including gravity, live.
Conclusion from the long digression below: yes, you can formulate gravity as a field in a flat manifold (e.g. Deser's, Doran-Gull-Lasenby, tele-parallelism...) but no, that manifold (and its additional structure) is not flat space-time. In any equivalent formulation of general relativity, you cannot avoid the coupling between the physical space-time metric and causal structure to the rest of the degrees of freedom (matter, forces...). Therefore, in any formulation, there are fields living on top of each other, and space-time amounts to relational coincidences between them. The underlying manifold serves as a necessary indexing device for the degrees of freedom of fields, but has no physical observable meaning. If the situation is such that the region of interest can be coordinatized by physical observables (e.g. matter fields embodying observers in that region), then we may use these as indexing the fields' coincidences; if moreover the coupling of those observables to gravity can be neglected, then they describe a physical Minkowskian space-time in that region. (But Minkowski's flat space-time is not a solution to Einstein's field equations when there is a non-vanishing cosmological constant! so assuming physical reality to a global flat 'space-time' is  meaningless in any equivalent formulation).

For a wonderful discussion and development of all these conceptual issues, you should read Carlo Rovelli's - Quantum Gravity, chapter 2 (in special 2.2, 2.3 and 2.4).
Einstein's general relativity is a theory about the dynamics of space-time; more precisely, it is the realization that the gravitational field 'is' space-time, and that non-relativistic (newtonian and minkowskian) space and time is a particular solution useful as background in regimes of approximately neglectable gravitational effects compared to the other phenomena of interests. As Einstein remarked in his original article, 
"... the requirement of general covariance takes away from space and time the last remnant of physical objectivity..."
A. Einstein, Grundlage der allgemeinen Relativitätstheorie, Ann. der Phys. 49 (1916) 769-822.
The equivalence principle along with general covariance, and determinism of a classical theory, requires the theory to be diffeomorphism invariant, i.e., to be background independent (see ref. therein). This can be understood by Einstein's hole argument. Observable space and time turn out to be purely relational notions, in the sense that the theory is formulated on a space-time manifold, but the same physics can be described by diffeomorphically equivalent manifolds, and ony the gravitational field in relation to the other fields on top of it define physical observables (therefore points of the auxiliary manifold lose any physical meaning, only spacetime coincidences of the fields define intrinsic relative positions). This is the transition from newtonian substantivalism to a purely relational universe where there are only dynamical entities "living on each other". From this Einstein's original point of view (and emphasized by relativists like those of canonical quantum gravity), talking about space-time or the gravitational field is the same thing, so the underlying $x^\mu$ coordinates of the manifold turn out to be a gauge artifact. This "general manifold coordinates" should not be confused with proper physical observables. Fixing a gauge is like choosing an appropriate manifold coordinate system where coordinates may or may not correspond to physical observables, and only afterwards we do calculate distances and times using the gravitational field (metric or tetrad) between events as specified by spacetime coincidences of the fields.
Therefore, asking about general relativity described as a field in a flat space (spacetime actually) is missing most of the conceptual revolutionary content of the theory itself. Nevertheless you can choose a particular solution, like minkowskian flat spacetime and expand the gravitational field as a perturbation on top of that. Then you get an effective field theory of the spin 2 perturbation on an unobservable flat spacetime with nonrenormalizable difficulties. This kind of approach is the original approach of the particle physics community through perturbative string theory. There is nothing wrong with this, but then one has to be metaphysically careful on what to call spacetime and gravitational field. Canonical/loop quantum gravity also suffers from serious difficulties, although it is background independent, so one should be open-minded as to what is the lesson to be learned from general relativity since we still lack a quantum theory of gravity.
In summary, from my personal point of view, and without a better experimentally successful theory of gravity than general relativity, we should commit ourselves to the "fact" that at least at the macroscopic, classical level, measurable space and time are relational concepts derived by the coupling of matter fields to the gravitational field where the underlying manifold serves as an indexing gauge with no physical meaning whatsoever. In this sense you can say that general relativity is already a field theory but on a diffeomorphism equivalence class of "underlying spacetimes" where the fields are defined, but you cannot talk about "flatness", thus no geometry only differential topology (and dimension and signature) because what is usually defined as flat about the geometry of physical space (and time) are the particular properties of physical trajectories and the relational geometry among the fields corresponding to euclidean/minkowskian properties.
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ADDITION: The physical content of diffeomorphism invariance is as follows. General covariance allows you to express physics in any reference frame (any spacetime curvilinear coordinate system), and that is useful to solve problems choosing the most appropriate coordinates. Once you have a solution of the field equations, you have your gravitational field (the metric or tetrad) and the equations of motion of other fields affected by that metric. Now, the solutions can be cast into any other general coordinate frame, possibly complicating the expressions due to the use of the best coordinates to get the solution. And here comes the magic: by the hole argument any diffeomorphism on a solution must be another solution, that is, our original solution in a new coordinate system must itself be a new solution in our original coordinate system!! This is because active diffeomorphisms (smoothly changing the underlying manifolds with the corresponding tranformation in the fields) and passive diffeomorphisms (coordinate changes) cannot be distinguished by background independence! The point is that the coordinates you choose during the whole process may not have any direct physical meaning (think of complicated coordinate changes to simplify the metric form to get the Schwarzschild or Friedmann solutions more easily). In particular, the physical distances and proper times are not those of the fancy coordinates in which your problem is easily solved, but need in general be expressed in terms of the metric, i.e. the gravitational field. So, given coordinates, you may talk about 2 points (events), their coordinate distance needs not be their spacetime interval, since the latter implies always the metric, and only in the flat Minkowski case you can identify both everywhere. In a general relativistic context, with nonvanishing gravity, there are no global flat (=inertial) coordinate frames, Riemann curvature is precisely the measure of that obstruction.
Therefore, in the way I understand the question, I do not see how to construct an underlying manifold with the same physical geometry in the case of nonflat spacetimes.
ADDITION+: concerning your second comment. Our universe spacetime is NOT flat, as it is expanding with acceleration. Minkowski's flat space-time is NOT a solution of the Einstein's field equations when there is a non-vanishing cosmological constant, thus the block universe cannot be globally flat. What is approximately flat on cosmological scales are the 3-dimensional SPACELIKE sections whose induced physical distance is expanding as seen by the comoving frame only defined at large scales like you want. But even in this solution, measured space distances are not coordinate distances, as they are related by the FRW scale factor which varies with cosmological time in that frame. So, do not confuse the spacelike 3-curvature (which may be flat) of a foliation of spacetime, with a global solution of spacetime. 
An offen confusing issue in relativity is that there is no absolute simultaneity. Even a cosmological solution like FRW, talks about "spaces that evolve in a global time", but that is just due to the convenience of the comoving frame, which is always an approximation for large scales. Every real observer is local and does not move with the cosmological comoving frame. It is important to remember the point that Einstein stressed: observable space and time are relational concepts defined operationally by measuring rulers and clocks. Calling "spacetime" to the underlying manifold is misleading. 
Gravitons and (quantum) field theory as we know them, are approximate theories of the world definable and only useful over regions of spacetime (not only space!) approximately flat for our purposes and precision. Fields, particles as their quanta, energy and similar concepts require the Poincaré symmetry group to be defined, and that is not available on the general relativistic case. That is why canonical/loop/spin quantum gravity is not defined in those terms, contrary to string theory. Everyday concepts of quantum field theory living on a general spacetime are misleading and only semiclassical; although classical field theory can be worked out, short range forces like the weak and strong nuclear forces are only quantum mechanical. So, even though general relativity is also an approximate macroscopic theory, its empirical adequacy and success teaches us that classical or quantum field theory on flat spacetime are also an approximate macro/microscopic model. As long as one is talking about the gravitational field in interaction with the other fields, both frameworks are not fully compatible, and that is the problem of quantum gravity.
ADDITION ABOUT DESER'S OR DORAN-GULL-LASENBY's WORK:, in other answers below and in a commentary of this one, it is said that the work by Deser or others 'shows' that general relativity can be for example described as a spin 2 field in flat Minkowskian space-time, and that this fact undermines my answer and misleads the reader to think that the answer to the question posted is 'no'. I would like to clarify how both answers are not only compatible and equivalently correct, but moreover prove the point that the confusions about the meaning of general relativity are the ones misleading learners of the theory due to a categorical error in ontology: the Minkowski space-time of special relativity is the set and causal structure of physical events measurable by observers in the approximation of negligible coupling between gravity and the observer's dynamics. On the other hand, the 'Minkowskian manifold' used in Deser's work and others is non-physical because it is not observable; THUS, what is misleading and confusing is calling "space-time" to an ingredient of an equivalent mathematical formulation which does not correlate operationally to the supposed physical meaning. Moreover, Deser's work shows again that the observable spatial and temporal properties of events are correlations between field coincidences of the gravitational spin-2 field and the rest, hence reinforcing Einstein's discovery that observable space-time is purely relational, whereas the underlying manifold structure is just a gauge artifact needed for indexing the degrees of freedom of these fields. The categorical mistake amounts to bestowing ontological weight to parts of the formalism which do not refer to the original entities. A physical theory consists of 3 elements: empirical facts, mathematical formalism and interpretation (the latter is the operational link between the former two). Newtonian and Minkowskian space-time are formalized respectively as an euclidean real affine space and a Lorentzian flat 4-manifold; their interpretation is the operational link between physical events and facts and their coordinates as referred to physical observers (in particular to inertial observers which are the smaller set of physical reference bodies whose state of motion leaves invariant the structure of the euclidean and minkowskian manifolds). This clearly allows us to understand the meaning of mechanics in a non-generally-covariant situation: observers measure distances and times, and at those points they measure local values of fields (like the electromagnetic field and a matter/charge distribution), then the equations of motion of Newton-Lagrange-Hamilton let them predict future correlations between field values at coordinated points of physical space-time. The difference with general relativity is that, in the general situation, there is a coupling between the degrees of freedom of any reference body (thus any observer) and gravity, so one cannot neglect the effects of dynamics on the observer, which means that the dynamics of matter and force fields affect on length, time and causal structure. Newtonian and Minkowskian space-time is assumed to be fixed and nondynamical, but the equivalence principle attributes the metric and causal structure to the effects of gravity, thus making any physical space-time subject to dynamics. One can formulate then a generally covariant theory where this insight is captured, either by the old pseudo-Riemannian metric on an 'underlying manifold', by a $SO(1,3)$ connection and tetrad of Lorentzian frames, or by any other manthematically equivalent structure. For example, in tele-parallel gravity one describes general relativity by purely flat 'underlying geometry' but where the degrees of freedom are captured by torsion. In Deser's formulation one describes gravity via a spin-2 field. BUT, the confusion is thinking that the underlying manifold structure, needed to formulate the theory, corresponds/can be interpreted as the physical structure! It is a historical accident that manifold coordinates $x^\mu$ have the same symbols than in the pre-generally covariant theories: the 'underlying manifold' $x^\mu$ do not refer to possible observer's measurements like in Newtonian and Minkowski's fixed space-times, and since the theory (Einstein's or Deser's) is completely diffeomorphic invariant, the physical events forming observable space-time are the relational coincidences of the fields living on top of each other. This is clearly emphasized by the mentioned formulation of Doran-Gull-Lasenby in their abstract:"In this manner all properties of the background spacetime are removed from physics, and what remains are a set of intrinsic relations between physical fields".
In particular, any splitting of gravity à la $g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}$, with $\eta_{\mu\nu}$ any solution of the field equations (Minkowski space in the usual case), separates the degrees of freedom of gravity if one maintains physical meaning to $\eta_{\mu\nu}$ because in that case one is describing a field $h_{\mu\nu}$ living on a particular solution of gravity, hence physical space-time, but restricted to the metric and causal structure given by just that solution; if like in Deser's work, one wishes to capture all the degrees of freedom in a spin-2 field, then the metric and causal structure of the underlying manifold no longer retain physical meaning. One should realize that whereas a physical theory may be underdetermined mathematically, its ontological commitments to which entities are real even in principle can only be challenged after theory refinements because of new empirical adequacy is needed. All equivalent formulations of a theory should not mislead us to think that there are different interpretations: the interpretation of Deser's (spin-2) formulation is obtained as rephrasing the usual one via its equivalence with the usual (metric/tetrad) formulation; the theory of general relativity was found out by careful analysis of previous physical principles, operational meaning to observability and mathematical consistency:
van Kampen’s 'Theorem': If you endow the mathematical symbols with more meaning than they operationally have, you yourself are responsible for the consequences, and you must not blame the original theory when you get into dire straits…
A: For above question, my answer is yes! Actually, for this question, a number of authors, including Rosen, Kraichnan, Gupta, Weinberg, Feynman, Deser, Grishchuk, T. M. Nieuwenhuizen, and Logunov, etc., have discussed the utility of introducing the metric of flat background space-time into GR. Those theories called the bi-metric gravitation or the field-theoretical approach to General Relativity. 
Generally speaking, they can be divided into four schools. I call them Rosen school, Weinberg school, Deser school and Logunov school. Rosen school uses the flat background space-time as a useful mathematical tool. Strictly speaking, the discription of Rosen school has mathematical problem, they use the same vector $dx^i$ to construct the metric of flat background space-time $ds^{2}=\eta_{ij}dx^{i}dx^{j}$ and the metric of Riemannian space-time $ds^{2}=g_{ij}dx^{i}dx^{j}$. If the $x^{i}$ is the coordinates of a flat space-time, it is impossible at same time to be the coordinates of the Riemannian space-time. If we think that the $g_{ij}$ is the metric tensor with respect to the coordinates $x^{i}$, here $x^{i}$ are not the coordinates of flat space-time anymore and its meaning has been totally changed. Weinberg school uses the flat background space-time in weak field of gravity for approximation calculus. That is ok, no mathematical problem. That is why now the mainstream thinks that considering gravitational problem in flat space-time is just for approximate calculus. Logunov school thinks that there is a effective Riemannian space-time in a real visible flat background space-time, they call their theory as Relativistic Theory of Gravitation. Their theory has same mathematical problem like Rosen school's and the picture of the effective Riemannian space-time and the real visible flat background space-time is not clear. Deser school has tried different analogs and guesses their Lagrangian densities according to the Einstein equations in order to corresponding with Einstein equations. I think these are very good works. Unfortunately, Deser school has not got rid of fetter of geometrization. They deduce Einstein equations from field concept in flat space-time and go back to GR. Deser said:"It is at this point that the geometrical interpretation of general relativity arises, since all matter now moves in an effective Riemann space of metric ..., and so the initial flat ‘background’ space ... is no longer observable". "It goes without saying that this non-geometrical interpretation of GR, far from replacing Einstein’s original geometrical vision, is a tribute to its scope". Here still has a similar mathematical problem like Rosen and Logunov schools. The gravitational field in flat space-time cannot be automatically converged to a Riemannian space-time through infinite series superposition. The coordinates $x^{i}$ in flat space-time cannot be changed to the coordinates of Riemannian space-time through infinite series superposition. There is no such mapping between a flat space-time and a Riemannian space-time. If no such mapping, what are the foundation and the geometrical significance that they guess and deduce Einstein equations? The reason they get these wrong ideas is that they do not know the relationship between Riemann geometry and the gravitational field in flat space-time. 
In my paper, the relationships of the Riemannian space-time, the de Donder conditions and the gravitational field in flat space-time has been discovered and elaborated. The main idea is: Let the coordinate system $x^{i}$ of flat space-time to absorb a second rank tensor field $Φ_{ij}$ of the flat space-time deforming into a Riemannian space-time, namely, the tensor field $Φ_{\mu\nu}$ is regarded as a metric tensor with respect to the coordinate system $x^{\mu}$. After done this, $x^{\mu}$ is not the coordinate system of flat space-time anymore, but is the coordinate system of the new Riemannian space-time. The inverse operation also can be done. According to these notions, we propose the concepts of the absorption operation and the desorption operation. Further, We find a way to hold the structural form of Einstein equations, abandon the purely geometrical interpretation of gravitation and make the equations global Lorentz invariance. Now Einstein equations have no any geometrical significance, and de Donder conditions have obvious physical meaning. Einstein equations and de Donder conditions together compose the equations of gravitational field in Minkowsky space-time. The gravitational red-shift of light is due to the photon obeying the conservation law of energy and the potential and kinetic energy transforming between each other when it is moving in a gravitational field. Gravitation can affect the motion of matter and change the energy-momentum of matter, thereby affect the period of matter’s motion, but these changes can be aware of by comparing with the background space-time. Just like usual, it is being routinely used in relativistic astrometry and relativistic celestial mechanics. People can store and analyze the data in terms of the flat space-time quantities after subtraction of the theoretically calculated gravitational corrections, rather than in terms of directly measured quantities.
For SCIRP, I have no many comments because this is my first time to publish paper there. They showed me the comments of peers review. Regarding the publishing fee, I find that many famous journals charge the publishing fee higher than them for open access.
Reference:
N. Rosen, “General Relativity and Flat Space I,” Physical Review, Vol. 57, No. 2, 1940. doi:10.1103/PhysRev.57147
S. Weinberg, “Derivation of Gauge Invariance and the Equivalence Principle from Lorentz Invariance of the S-Matrix,” Physics Letters, Vol. 9, No. 4, 1964. doi:10.1016/0031-9163(64)90396-8
S. Deser, “Self-Interaction and Gauge Invariance,” Gene- ral Relativity and Gravitation, Vol. 1, No.18. doi:10.1007/BF00759198
S. V. Babak and L. P. Grishchuk, “Energy-Momentum Tensor for the Gravitational Field,” Physical Review D, Vol. 61, No. 2, 1999, Article ID: 024038. doi:10.1103/PhysRevD.61.024038
A. A. Logunov, “The Relativistic Theory of Gravitation,” Nauka, Moscow, 2000.
G. Liu, "Riemannian Space-Time, de Donder Conditions and Gravitational Field in Flat Space-Time," International Journal of Astronomy and Astrophysics, Vol. 3 No. 1, 2013, pp. 8-19. doi: 10.4236/ijaa.2013.31002.
