Many problems must be solved before they can exist.

(1) Only suitable material for the cable is, at the moment, unobtanium. (2) Cable must be moved continuously to dodge debris and satellites whose orbits cross the equator (all of them except those in geosynchronous orbit). (3) The cable, elevator module and any cargo or people must pass through the Van Allen belts, which will degrade them. People don't respond well to degradation by radiation.

Other than that, we're good to go.