How can mass bend spacetime, if there's nothing to bend? The Michelson-Morley experiment proved there was no aether, nothing that light moves through in space. Yet in GR, mass bends spacetime so that light travels in arcs around large masses. How can spacetime be bent, if there is nothing there to bend?
This question is not related to the factors that bend spacetime, which I noticed many physicists try to shuck it off to, to avoid giving a coherent answer, but the structure of spacetime itself.
 A: To answer your question, we first need to understand exactly what is mean by the notion of "bend" when physicists use it to describe the "warping" or "curvature" of spacetime. Then with a more precise understanding of this notion, we can begin to address your question proper. In summary:

*

*You shouldn't expect your everyday intuitive notion of bending and curvature of familiar everyday objects to hold accurately when thinking of "spacetime bending".

*Even so, it is simply "spacetime" itself that bends, and spacetime is not the same as "nothing". Right now, we (physics) does not know anything more than this statement. We simply know, through General Relativity and many supporting astronomical observations, that the geometric properties of even "empty" spacetime do vary from place to place and that therefore space and time does have a separate "material" (in the sense of physical) reality from the rest of the "stuff" that makes up our physical world. We could change your question from "what is it that bends?" to "what is the underlying mechanism within space and time itself that leads to geometrical, measurable variations in spacetime properties?". The answer to this one is that we simply don't know yet, at least there are no generally accepted theories. The quest for the answer to this latter question is roughly what the quest for a theory of quantum gravity is all about.

*This separate physical reality is in no way contrary to the falsification of the 19th century notion of "Aether" through the Michelson-Morley and similar experiments.

Now, some more details on these points.

1 The Correct Notion of "Bending"
The statement "stress energy bends/ warps / curves spacetime" can lead to conceptual problems if you try to read it in a too "everyday" way. The "bending" or "curvature" notion here is related to the "bentness" or "warpedness" that we ascribe to everyday objects only at the most abstract of levels.
The correct abstraction to make for the purposes of General Relativity is a systematic, objective way of measuring how much the geometry of the world around us deviates from Euclidean postulates (or a analogy that takes special relativity into account - Lorentzian postulates). An object called the Riemann curvature tensor does this: this object vanishes in a region if and only the geometry of spacetime is the same as that of special relativity. This object makes its quantification by computing how much a vector is shifted by being parallel transported around a loop.
In "flat" spacetime, parallel transport around any loop does not lead to any change in a vector. This is the definition of flatness, and its negation is the definition of curvature. There is no more, nor any less, to the notion of curvature than this: in particular, you should not expect to be able to visualize it.
The fact that experimentally we do indeed find that this object measures a non-flatness in some regions in spacetime and not others shows very forcibly that spacetime is a real, "material" (in a Lorentz-invariant) way; "empty" space is not nothing and can have properties that differ for different regions of "empty" space depending on whether stress energy or gravitational waves are present and in what intensities.
If this notion  of curvature seems a little abstract, then understand that like most mathematical notions with roots in the everyday world, it has a very long history of formulation to account for very subtle effects, beginning most notably with the work of János Bolyai; see also this answer of mine.
This notion also specializes to aspects of our everyday intuition in many, surprising, fascinating and intellectually fulfilling ways:

*

*Angles in triangles drawn in the spatial part of spacetime sum up to 180 degrees if and only if the curvature tensor vanishes in the region in question. To make this notion meaningful, one must be able to slice spacetime up into "now" slices of constant time; this too is always possible if the curvature tensor vanishes in a region. Check this on the surface of a sphere, e.g. for a triangle comprising geodesics joining the North Pole and two equatorial points. Parallel transport around such a loop also turns a vector through an angle equal to the angle at the North Pole's vertex.


*The Euclid parallel postulate in the spatial part of spacetime holds if and only if the curvature tensor vanishes in the region in question. Again, one must be able to slice spacetime up as described above: this can always be done if the curvature tensor vanishes (but there are some non-flat spacetimes where we can still do such slicing).


*A co-ordinate transformation exists to a system of flat, Minkowski co-ordinates in a region if and only if the curvature tensor vanishes throughout that region. The curvature  tensor is sometimes said to measure the "blockage" that hinders the integrability of equations that describe an attempt at such a transformation.

2 What is it that Bends in this Sense?
Okay, so we've established that spacetime bending is only vaguely like the bending of fabric at the most abstract of levels. Even so, there is something about empty spacetime that leads to measurable differences in its properties, depending on what stress energy or gravitational waves are in the neighborhood of a spacetime region. So space and time seem to have a separate physical reality, and you are asking more about the nature of that reality.
The simple answer is that physics doesn't have a good answer to this question as yet - the quest for this answer is roughly what quantum gravity is about.
Notwithstanding general relativity's power and sophistication, we must take a step back and understand that it is ultimately a very simple theory. It describes spacetime geometrical properties and how they change under the influence of stress energy and gravitational waves, but it says nothing about the underlying mechanism.
3 So What of Aethers and the Michelson Morley Experiment?
So, spacetime has a separate physical reality, with properties that can change; measurably so, albeit very subtly so aside from in extreme situations that do not arise in our everyday lives. What's the difference, then, between this and an Aether?

*

*In fact, the Michelson Morley experiment does not falsify an Aether. It simply rules out any Aether that does not fulfill Galileo's relativity principle. It tends to be forgotten that Lorentz Aether Theory is fully consistent with the MM experiment. Lorentz Aether theory was cast aside not because it was wrong - its predictions were exactly the same as those of special relativity - but because it was more unwieldy than special relativity and ultimately SR overtook it because SR could be readily generalized to general relativity. I say more about this in my answer here; see also ACuriousMind's exposition here.


*Like the Aether of Lorentz Aether theory, all of the subtle curvature effects are Lorentz invariant. They simply will not be measured by anything like the Michelson Morley experiment. They are so-called "tidal" or second order effects: they are detected by comparing measurements made at separated points in spacetime (e.g. one has to transport a vector around a loop of nonzero extent to perceive them). The MM experiment was a local experiment comparing measurements made at the same point in spacetime. The first experiment to compare measurements at different points in this way was the Pound Rebka experiment and it took a great deal of ingenuity to make the comparison between two separated points (on different floors of Harvard University's Jefferson Laboratory) because the effects over that separation are so tiny. The Hafele-Keating also made such a comparison, but was not as "clean" as it was influences by two different relativistic effects, one of the simple relativistic time dilation.
So, in the sense it can have properties differing from place to place, spacetime is a kind of medium. However, whatever the ultimate nature of spacetime turns out to be in some future quantum gravity theory, it will be nothing like most of the 19th century conceptions of an Aether and of course will be in keeping with all experimental observations, the Michelson-Morley results being only one of many.
A: EDIT 
I wrote this reply last night because I think the OP is asking a philosophical question, akin to What is an electron? I then deleted it, as a personal opinion on philosophical question.
I fully accept that the other descriptions are correct, because they have been tested to the limits of our measuring ability. But  I still think an answer to this question based around coordinate systems and the differential geometry of a 4 D manifold, will eventually need to be modified if we find:


*

*that space is discrete or

*we are required to modify our current approach, to achieve a quantum theory of gravity.
In this way, we might eventually put the current philosophical question of what "is" space, into a  mathematical form that  acceptable to the physics community.
END EDIT
I think the other answers, apart from answer above  (13526) in the comments, all unintentionally  do not fully answer  your question, because you are asking physicists to answer a philosophical question.
If I go downtown to have a coffee, taking 5 minutes to walk the mile to the cafe, that is a spacetime distance on  a local scale, and it is the the way we think about distances (intervals in GR). And the same idea of space as a geometrical / coordinate system distance applies to as far we can observe looking outwards. 
The problem arises when we try to view spacetime on a universal scale, when we don't have edges to the universe, (no matter whether K is equal to 1, greater  than 1 or less than 1, that is that spacetime  is either  flat, closed or open). 
We now have to find a different definition of spacetime because the local definition will not make any sense. The universe did not start at a point, it has no centre and it has no edges. Look up the number of times these questions have been asked here already.
So on a global scale, we (in my opinion), need to redefine spacetime as the relationship between objects in the universe and reconsider the idea that  geometry and continuous 4D differentiable manifolds are the complete solution to the spacetime of the observable universe. Geometry may be of no use when describing an electron, and the beginning of the universe was "smaller" than that. 
It is also not useful in describing the interior of black holes or if we discover that spacetime is actually discrete. (A long shot, I appreciate that).
So at both scales of the universe, at its "smallest", and at its "biggest", (remember we are in an expanding universe), the descriptions of the other answers of what space "is" fail us, they simply do not apply.
In summary, on a local scale, I believe that the geometry and coordinate system, while you may consider that they  simply  replace one word for another and avoid telling you what spacetime really "is", (because that is not the job of physics in the first place), they  are essential if we are to keep track of our measurements on a local scale.
But, on a global / universal spacetime scale, I don't believe they deal fully in describing the observable universe from its "tiny" initial state, to its expanding edge state today. 
A caveat here is that a closed universe may "kinda" have an edge, in that if you start at any point you may return to it, but the evidence points to a flat but expanding universe.
So I ask you to consider you a description of spacetime on a global scale as a set of relationships. I think this is more appropriate, in fact the only way, we can deal with what spacetime is, on a universal scale.
Using the handwavy word "relationship" brings in QM, which will need to be brought in anyway if we want to achieve  a quantum gravity theory,   but I have now moved from a personal opinion, to complete uninformed  speculation. Time to stop.
My sincere apologies to those concerned, for the number of edits in this post, I will not repeat this practice in future.
A: Nobel laureate Robert B. Laughlin has some interesting things to say about general relativity and ether. I'll quote from his book "A Different Universe". Laughlin is discontent with the strategies and underlying dogmas of modern physics; in the book he tries to provoke us out of our comfort zone. 
The book's gist is that modern physics, focusing on elementary particles and increasingly smaller and detached details, misses the point: The emergent phenomena whose underlying building blocks are secondary — much like it is secondary whether a house is built from brick or granite blocks. With respect to gravity he argues that dogmatically insisting that there is no ether makes us miss the emergent properties of space time. Acknowledging and exploring those would be more relevant and productive than trying to pinpoint the increasingly narrow and irrelevant details on the elementary level, a strategy which  he dismisses as reductionist. Pp. 120 ff, Insertions and bold emphasis are by me:

The equations Einstein proposed to describe gravity are similar to those of an elastic medium, such as a sheet of rubber. [...] Conventional [static] gravity is thus like the dimples under the feet of a water skimmer, and gravitational radiation [e.g. from rotating binaries] is like the disturbances generated by the skimmer when he scampers away. [...]
The irony is that Einstein's most creative work, the general theory of relativity, should boil down to conceptualize space as a medium when his original premise was that no such medium existed. [The ancient Greek and Maxwell assumed that there was an ether.] Einstein, in contrast, utterly rejected the idea of ether and inferred from its nonexistence that the equations of electromagnetism had to be relative. But this same thought process led in the end to the very ether that he had first rejected, albeit one with some special properties that ordinary elastic matter does not have.
The word "ether" has extremely negative connotations in theoretical physics because of its past association with opposition to relativity. This is unfortunate because, stripped of these connections, it rather nicely captures the way most physicists actually think about the vacuum. [...]
[Space] is filled with "stuff" that is normally transparent but can be made visible by hitting it sufficiently hard to knock out a part. The modern concept of the vacuum of space, confirmed every day by experiment, is a relativistic ether. But we do not call it this because it is taboo. [...]
The clash between the philosophy of general relativity and what the theory actually says has never been reconciled by physicists and sometimes gives the subject a Kafkaesque flavor. On the on hand, we have the view, founded in the success of relativity, that space is something fundamentally different from the matter moving in it and thus is not understandable through analogy with ordinary things. On the other, we have the obvious similarities between Einstein's gravity and the dynamic warping of real surfaces, leading us to describe space-time as fabric. Bright young students inevitably pick up on this and ask the professor what moves when gravitational radiation propagates. They receive the answer that space-time itself does, which stops them cold. It is like learning that the surface of the sea undulates because it is an undulating surface.1 Wise students do not ask this question a second time.

Keep asking ;-).

1 In a footnote Laughlin quotes Moliere who lampoons a group of physicians providing an explanation of the sleep-inducing properties of opium as stemming from its "virtus dormitiva" [dormative power]. See this wiktionary entry.
A: There is spacetime to bend. The understanding of spacetime becomes vital to understand bending. One can comprehend space easily but when time is added as a fourth dimension, it becomes difficult to visualize such a situation. Its a 4 dimensional fabric.
A: Your basic foundations of the understanding of General Relativity are not clear, it is not related to the ether, but it is related to the space-time fabric. Space and time do exist into a single four dimensional continuum, this four dimensional fabric is warped by the presence of mass or energy on it. Here is a simplified explanation :-
The first principle Einstein devised was the Principle of Equivalence, which stated that Gravity and Acceleration are the same thing, what he meant to say was that if you are travelling in an elevator in an upward direction with an acceleration that is equal to g = 9.8 m/s^2 and if you are standing stationary on the surface of Earth, and the Earth is exerting a gravitational pull on you downwards with an acceleration that is equal to g = 9.8 m/s^2 then you would not be able to differentiate between the two situations, you would feel exactly the same being in an elevator going up with an acceleration of g or standing on earth with a gravitational pull of acceleration of g, and so since it is now proven that Gravity and Acceleration are one and the same thing we can now see why Gravity bends light.
Going back in the elevator again, let us suppose the elevator is travelling upward again with a certain acceleration (A ) and now we take a torch and emit a light beam from one wall of the elevator to the other, and since light travels at a constant speed :- C we would observe that by the time the light beam reaches the other side of the Elevator, the Elevator would have moved up, since it is accelerating up, and so if we trace the path of the light it would be curved. Therefore, it is true that light is bent in an accelerating body, but we have also proved that Acceleration and Gravity are the same phenomenon. Therefore, Gravity must also bend light passing near it.
Now the last concept is that how can mass warp space and time, very simple. If we were to plot the motion of an accelerating body on graph with the X and Y axes of Space and Time, we would represent it as a curved line on the graph, since its velocity is variable. Now we know gravity causes acceleration, and it makes every body accelerate whether it be at rest or in motion, therefore in the first case we plotted a curved line because the body seldom was accelerating, but in the second case if we were to plot the graph of acceleration caused due to gravity, we would represent the space and time axes as curved and the line of motion of the body as straight, and therefore it is seen that mass curves space and time.
A: Matter bends space and time itself. This is not related to an aether
