Our current technologies are unable to provide materials that would be strong enough to build a space elevator on Earth, the gravity here is just too strong, though carbon nanotubes have proved strong enough to fullfil the theoretical requirements. But imagine what happens when you have two (or more) bar magnets. If you place the two "north" poles of the bar magnets together, they try to repel each other. If we made a space elevator piecemeal, using alternately repelling bar magnet principles, would some of the weight of our space elevator be taken on by the electromagnetic field, meaning that we could actually build a space elevator with the materials and technologies we currently have? We could power these magnets using space solar power attached to the end of our elevator. For an alternative solution, see manipulated spacetimes.
See isec.org for more information on the space elevator.