I guess the biggest problem has always been to find a material that is strong enough and light enough that it could support it's own weight over the kind of distance involved.

Now they are working with carbon nanotubes --- microscopic tubes of carbon that have a single molecular structure. They are incredibly strong and light, and theoretically they could work as tether material. There are processes for making small amounts of the nanotubes, but no good maufacturing processes yet for actually making them into a tether. That is one technical hurdle that is probably not very far away from being solved.