If the force of gravity can be thought of as masses leaving dents on a sheet of spacetime, what is holding up that sheet?
The visual is a bit misleading. There is no “up” in spacetime and there is no need for an additional force pulling down. More importantly, the visual neglects any curvature along the time direction, which is essential for understanding gravity.
Step back from the visual for a moment. Consider an ant walking slowly on the surface of a ball. Let’s say that this ant never turns but goes straight forward every step. Even though the ant walks straight it will eventually wind up back where it started and will have traveled along a great circle.
This is the meaning of curved spacetime. It isn’t about some force outside of spacetime pulling things down, like in the sheet analogy. Instead, it is about the fact that if you always travel straight ahead on a curved surface your path can do weird things like loop around or otherwise curve from a different perspective.
Nothing. There is no sheet. It's just a metaphor to help you visualise the bending of three-dimensional space. Metaphors always break down if you push them too far...