How to write down various metrics without coordinates? For example Schwarzschild metric, or Alcubierre metric, but using only intrinsic (natural, canonical, etc.) physical objects (like length,  angles, etc.) for relations between natural objects on tangent structures. So not using anything like basis vectors, frames, charts, etc.
A metric tensor is just function of angles and absolute values of vectors. It is a section of the dual of the tensor product bundle of tangent bundle with itself. It has nothing to do with coordinates. So why can't it be written down without them?
 A: I'm not sure if this will precisely answer your question concerning "metrics".... but this might have the spirit of
what you may be seeking.
Here's an overview of a coordinate-free derivation of the Schwarzschild solution by Robert Geroch.
[short answer: using symmetries specified by Killing vector fields, construct various scalar fields for use in the Einstein Fields Equations to obtain a set of differential equations for those scalar fields. After the solutions are obtained,
the results can be expressed in coordinate-form, if desired.]
(sources:
General Relativity: 1972 Lecture Notes (Lecture Notes Series) (Volume 1)
Minkowski Institute Press; 1 edition (February 25, 2013)  ISBN 978-0987987174

also
http://home.uchicago.edu/~geroch/Course%20Notes (latexed draft?)

http://www.gravity.psu.edu/links/general_relativity_notes.pdf (scan of original notes)     )
Refer to the above for details.
Below I will quote some passages from the LaTeXed file and summarize some parts of the approach given. (Hopefully my transcriptions are accurate.)

Ch 25: The Schwarzschild Solution
Physically, the Schwarzschild solution represents the geometry of an
  “isolated, non rotating star, which has settled down to equilibrium”.
  What properties would we expect such a solution to have? Firstly,
  we would expect the solution to be static, i.e., we would expect to
  have a timelike, hypersurface-orthogonal Killing vector $t^a$. Secondly,
  we would expect the solution to be spherically symmetric, i.e., we
  would expect to have Killing vectors ${l_1}^a$, ${l_2}^a$, ${l_3}^a$
  which are spacelike,
  linearly dependent, and have the commutation relations 
  $$[{l_1},{l_2}]^a={l_3}^a\quad [{l_2},{l_3}]^a={l_1}^a\quad [{l_3},{l_1}]^a={l_2}^a\quad (79)$$
  Finally,
  we would expect that the time-translations and rotations commute,
  i.e., we would expect to have additional commutation relations
  $$[t,{l_1}]^a=[t,{l_2}]^a=[t,{l_3}]^a=0\quad (80)$$
  To summarize, we are concerned with space-time having four Killing
  vectors, with the commutation relations (79) and (80). For the matter
  composing the star, we take a fluid. Thus, we have the density $\rho$,
  pressure $p$, and (unit) velocity field $\eta^a$. 
  Since the star is supposed to have “settled down to equilibrium”, we suppose that the fluid does not
  “move relative to static observers”, i,e., 
  we take $\eta^a$ a multiple of $t^a$.
  
  To summarize, the Schwarzschild solution is a space-time with four
  Killing vectors, $t^a$ (timelike, hypersurface-orthogonal), and 
  ${l_1}^a$, ${l_2}^a$, ${l_3}^a$
  (spacelike, linearly dependent), subject to (79) and (80), where the matter
  is a fluid with four-velocity field proportional to $t^a$. We now discuss
  the geometry of the Schwarzschild solution.

Then, Geroch proceeds as follows:


*

*Define a scalar field $\lambda=t^a t_a$. ($\lambda<0$ since $t^a$ is timelike [signature $(-+++)$])

*Write Ricci in terms of $\lambda$ using the hypersurface-orthogonality of $t^a$:
$$R_{mb} t^m =\frac{1}{2}\lambda^{-2}t_b(\nabla^c \lambda \nabla_c \lambda) -\frac{1}{2}\lambda^{-1}t_b\nabla^2\lambda\quad (83)$$ 

*Use the Einstein field equations for a perfect fluid to introduce matter variables (in place of the Ricci terms) to obtain
$$
R_{ab}=8\pi G\left[ 
-\lambda^{-1}(\rho+p)t_a t_b+\frac{1}{2}(\rho-p)g_{ab}\right] \quad(84)
$$
$$
\lambda^{-1}\nabla^2\lambda-\lambda^{-2}(\nabla^c \lambda \nabla_c \lambda)
=8\pi G(\rho+ 3p)
\quad (85) $$
which "can be rewritten in the more suggestive form"
$$\nabla^2 \left[\frac{1}{2}\ln(-\lambda)\right]=4\pi G(\rho+3p)
\quad (86)$$


*Define a positive scalar field $r$ in spacetime as
$$2r^2=
l_1{}^a l_1{}_a
+
l_2{}^a l_2{}_a
+l_3{}^a l_3{}_a\qquad (88) $$ 
which he describes "as a sort of 'radial distance from the center of the star' "

*Define the scalar field 
$\mu=(\nabla^a r)\nabla_a r$, where
$\mu=1$ for flat space, and deviations of $\mu$ from 1 represent the "curvature of space"

Let us summarize the situation. We think of $r$ as a “radial coordinate”.
  We think of $\lambda$ and $\mu$ as "fields which describe the geometry
  of space-time." 
  Since our space-time is static and spherically symmetric,
  we expect that everything of interest will be a function only of $r$....
  

  The idea is to use Einstein’s equation to obtain a pair of ordinary differential equations on the functions $\lambda(r)$ 
  and $\mu(r)$.

Eventually, for the region outside the star (so $\rho=0$, $p=0$), Geroch arrives at these
$$\lambda''\mu -\frac{1}{2}\lambda^{-1} \mu(\lambda')^2+
\frac{1}{2}\lambda'\mu' +2\mu r^{-1} \lambda'=0\quad(94)$$
$$-\frac{1}{4}\lambda^{-1} \mu \lambda' \mu'
-\mu \mu' r^{-1} + \frac{1}{4}\lambda^{-2} \mu^2(\lambda')^2
-\frac{1}{2}\lambda^{-1} \mu^2\lambda''=0\quad(95)$$
where $d/dr$ is denoted by a prime.

We have now obtained the ordinary differential equations we sought. What remains is to solve them.
  Eliminating $\lambda''$ between (94) and (95),
  we obtain simply $\lambda'/\lambda=\mu'/\mu$. 
  So, $\lambda$ is a constant multiple of $\mu$. 
  What multiple should we choose?
  
$\vdots$

  [physical and mathematical arguments]
  In Minkowski space, $\lambda=-1$ and $\mu=1$, which suggests $\lambda=-\mu$.
  
$\vdots$

  Setting $\lambda=-\mu$ in (95)... the solution is $\lambda=a+b/r$

$\vdots$

  We write $\lambda= −1+2GM/r$...
  
$\vdots$


  It should now be clear that one can choose coordinates in which
  the metric for the Schwarzschild solution takes the well-known form
  $$−\left(1 −\frac{2GM}{r} \right) dt^2 + 
\left(1 −\frac{2GM}{r}\right)^{−1} dr^2 + 
r^2( d\theta^2 + sin^2 \theta d\phi^2)$$
  The $\theta$ and $\phi$ are "angular coordinates", while the scalar field $r$ becomes a "radial coordinate".

...so coordinates are introduced at the last step.
A: If I understand your question properly, this is not generally possible.  Take Schwarzschild, for example, which is spherically symmetric.  You have one "special" point at the singularity, but you have nothing else physical to which you could make reference for specifying the metric.  (Depending on your view of your goal, you might say the event horizon is also a "physical" surface, but even with that you cannot specify angles relative to anything special in the spacetime.)  Your only choice is to introduce some structure of your own for writing it down, which is typically a coordinate system.
If you have a spacetime with more structure, it might or might not be possible depending on the structure.  There's no way to answer exactly how to do it at this level of generality.
A: An interesting approach that achieves what OP is after in geometries of physical importance is detailed in a series of papers by Vasiliev and collaborators [1–3]. There a series of asymptotically AdS black hole solutions (Kerr–AdS${}_4$ and its various generalizations) are given a coordinate-free descriptions as a deformation of a background anti-de Sitter geometry with a distinguished Killing vector (with the parameter of deformation in the simplest case being the mass of the black hole, and the spin parameter determined by the kinematics of the Killing vector).
An important feature of the construction is that metrics belong to the Kerr–Schild class, which means that the Kerr–Schild vector is null and geodesic vector of both the background and deformed geometries.


*

*Didenko, V. E., Matveev, A. S., & Vasiliev, M. A. (2008). Unfolded description of AdS4 Kerr black hole. Physics Letters B, 665(4), 284-293, DOI:10.1016/j.physletb.2008.05.067, 
arXiv:0801.2213.

*Didenko, V. E., Matveev, A. S., & Vasiliev, M. A. (2009). Unfolded dynamics and parameter flow of generic AdS(4) black hole, arXiv:0901.2172.

*Didenko, V. E. (2011). Coordinate independent approach to 5D black holes. Classical and Quantum Gravity, 29(2), 025009, doi:10.1088/0264-9381/29/2/025009, arXiv:1108.4321.
