Naive.

How would a fabric of nanotubes be like?

This is something everyone did once in their lives, just not with nanotubes:

Stack some nanotubes side by side. Now get other nanotubes and orient them orthogonal to the first set of tubes. Now pass the first tube in this new set over the first tube of the other set, under the second tube, over the third, and so on, alternating. 

The next tube you will pass, you should first pass it under, then over, then under the tubes, and so on. Repeat until all tubes are passed. You end up with a fabric, with one tube supporting the other.