It's entirely possible that in a few hundred years or so the answer below will be known to be wrong but at the moment it's better to think of "the fabric of spacetime" as a matter of convenience, rather than actually being some kind of stuff.
The main reason for this is because if spacetime actually has its own substance then we are unlikely ever to be in a position to notice ourselves. The main place we might see that it is stuff is inside a black Hole, but unfortunately they have an annoying tendency to kill you before you could tell anyone. The scale, at any rate, has to be ludicrously small -- we're talking distance or time intervals measured in trillionths of trillionths of trillionths of a metre, and a similarly tiny fraction of a second. To probe that kind of scale requires basically too much energy (or, more importantly, money), at least for the foreseeable future, for it to be realistic to expect that we could test.
Still, we can hope that maybe we'll understand it one day. In the meantime, the best theory of spacetime we have, ie General Relativity, is a bit dishonest. Space and time aren't really "things" so much as "convenient points of reference to describe where you are in the Universe".