You can get started with some rubber bands and soda straws. You just need a lot of patience. That will get you started but there are a limited number of things you can do with that and the rubber bands won't last being under constant tension and will eventually break. It takes a surprisingly small amount of materials, just something for the struts like dowels and some string. The most difficult thing is choosing how to attach the strings to the rods. It takes a lot of adjustment to get the pretensioning correct so you need something that you can change until you get it where you want it.
For something like a rubber band the challenge is that it will tend to pull itself apart before you get the final band attached. For cables that don't stretch the challenge will be it will be a gigantic tangled mess until you tension it. I have no idea how Snelson managed to construct the larger sculptures without killing himself.
Do a search for "tensegrity rods two-nails fuller mailing list" seems to produce a fair number of hits.
Also see:
An Introduction to Tensegrity by Anthony Pugh, LOC: TA658.2 P85x, copyright 1976, University of California Press, ISDN: 0-520-02996-8 (cloth/hard) or 0-520-03055-9 (paper), 121pp.