That's exactly like asking, "What is time made of?"
The simplest answer from a physics perspective is that space, like time, is just a measure of position.
Imagine if our universe was actually a giant computer simulation, like a 3D computer game only much more complicated. In a typical online 3D computer game, you might throw an object or shoot a bullet. The computer keeps track of the virtual object or bullet as it travels through virtual space. Of course, the game is imaginary: there is no bullet nor any space for it to travel through. The "bullet" is a bit of data in the computer's memory and its position in virtual space is stored as a series of numbers.
There are some valid reasons to believe that our universe may actually be something like the game described above, with subatomic particles being analogous to the "virtual bullets." It's an emerging scientific concept known as the "holographic principle."
So, would you ask what the virtual "space" inside a computer game was made of? Perhaps the question is just as meaningless for our "real" universe.
There's another way to look at your question: You might be wondering, "How can there be a region of space with nothing inside it?" In fact, under the standard understanding of physics as it exists today things like gravitational fields and electromagnetic waves extended forever and inhabit every possible region of space. Furthermore, since gravitation force is thought be carried by "gravitons" and electromagnetic force is carried by photons it must be that some gravitons and some photons exist everywhere in space. Thus no region of space is truly "empty."
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