Space Tethers.
You use'em to slingshot things into orbit, or maybe out of orbit too. The problem is that the cable has to be really strong, and there's a lot of crap floating around up there that will sever a cable. One solution to the breakage problem is to make a cable into a kind of mesh, so breaks don't matter as much.
Making the tether out of a mesh is a pretty cool idea, but all you've done is extend the lifetime by some factor. What you really want to do is find a way to repair the damage relatively easily.
Picture two mesh tethers between the endpoints. Each tether is made of a series of lines. The lines come out of the tether and are _unwoven_ from the mesh weave. They are then looped back around and _reweaved_ into the tether going back in the other direction. Each line within a tether is actually participating in a complete loop, there are back again. Each line is an unbroken circle.
The tether is then _moved_ through the continuous loop, unweaving and weaving at each end. In this way the tether acts like a belt.
If a break occurs, then movement of the belt/tether will eventually bring the break to one of the terminals, where it can be repaired. The weave localizes the damage and ensures that the line will not simply fly off into space. The repaired line is then rewoven into the loop.
A belt-like tether like this can last indefinitely.
1:51:58 AM 
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