I would absolutely love to work on a space elevator; maybe when I finish up school I'll have the chance. Meanwhile, I have been emailing some of the groups involved and trying to scope out the scene, and I think that the project concept is entirely realistic, even if it sounds like science fiction.
The basic idea is simple: put a large satellite or space station in geosynchronous orbit, and then drop a cable down to the surface of the earth. Since the satellite would be stationary relative to the planet below, the cable would hang motionless in the atmosphere and could be used by an elevator system to lift cargo and people up into orbit at a tiny fraction of the cost that current rocket technologies require. The science needed to make this concept a reality already exists, and LiftPort Inc. (and the related company, HighLift Systems) intends to follow through and build the thing within 15 years.
Rather than rehash what has already been written, please take a look at the links below:
How Space Elevators Will Work.
LiftPrt Inc., a company created to build a space elevator within 15 years.
Space.com article, "The Space Elevator Comes Closer to Reality".
Wired article, "To the Moon in a Space Elevator?"