It's more complicated than that. The curvature of a 4-dimensional manifold can't be described by a single number, the way the curvature of a 2-d surface can. In general it takes 20 numbers for a 4-d manifold; there are cases with special symmetries where fewer numbers than that are sufficient, but it's never just one number.
The other thing to keep in mind is that the curvature is of
spacetime, not just space. So one of the dimensions of the fabric is the time dimension. Most visualizations, such as the "ball denting the rubber sheet" one, or even the "wormhole" one, are only showing you the curvature of space, not spacetime. But much of the important information about spacetime curvature requires looking at time, so it's being left out in those visualizations.
Instead of trying to visualize the "fabric" all in one go, another avenue that is open is to think of what spacetime curvature means, physically. It's often equated to "gravity", but it's really more specific than that: it's
tidal gravity. For example, suppose I have two rocks hanging motionless above the Earth at some instant; both of them lie along a single radial line from the center of the Earth, but one is a little bit higher than the other. As the rocks start to freely fall, their separation will increase (because, in Newtonian terms, the lower one will fall slightly faster than the higher one). This is an example of tidal gravity. Or, if we have two rocks that are both at the same altitude but separated a little bit horizontally, and they start to fall, their separation will decrease (because, in Newtonian terms, they are both falling towards the center of the Earth, and that is a slightly different direction for the two of them). That is also an example of tidal gravity.
If we now try to translate what I just described into a visualization, we will find that it can't be done with a single "sheet" of fabric, so to speak--at least, not in any way that will be easy to interpret. But we can do it with two "sheets". For one "sheet", we have the time direction along one axis, and the radial (vertical) direction along the other; and we draw the worldlines (paths through spacetime) of the two radially separated rocks on the sheet. We find that the rocks separate, which means the "grid lines" on the sheet must get further apart in the radial direction as we move along the time direction. This means the sheet will be shaped something like a saddle.
For the other "sheet", we have the time direction along one axis, and the tangential (horizontal) direction along the other, and we draw the worldlines of the two rocks. We find that they get closer together, which means that the "grid lines" on the sheet get closer together as we move along the time direction. This means the sheet will be shaped something like a section of a sphere.
For an image of what I'm talking about, look at page 112 here:
https://books.google.com/books?id=iGmPBAAAQBAJ&pg=PA111&lpg=PA111&dq=tidal+gravity+spacetime&source=bl&ots=wuZVgeCIup&sig=klK7uzpV6hLeG2eRCv2MeUtNvKk&hl=en&sa=X&ei=-u1cVYCUIYaWNt_sgHg&ved=0CCsQ6AEwAjgK#v=onepage&q=tidal gravity spacetime&f=false
This is from Kip Thorne's
Black Holes and Time Warps, which I highly recommend as a book on GR for the non-technical reader. The text is talking about the tidal gravity produced by the Moon (so the "rocks" would instead be pieces of Earth's ocean), but the tidal gravity produced by the Earth, or indeed any spherical gravitating body, is similar.
