http://www.cosmosfrontier.com/files/SpaceElevator.JPG
Use our drilling tech to anchor and build the tallest structure in the world (probably around 10km high), the elevator will need to be able to move up and down, and the top of the tower will need to be able to bend a little, but still have guy wires... probably use some electromagnetic shock absorbers to counteract wind loading and such. Find an asteroid and figure out a way to maneuver it into geosynchronous orbit. Build a solar or small nuclear powered gyroscope to move the asteroid to ensure proper tension and position is maintained. Launch sections of carbon nanotube wire that can be combined in space and lowered. Then build the electromagnetic track going up.
Filling a hydrogen balloon and lifting the strands up might also work to get out of the atmosphere, but it's the remaining 35,000+km and dealing with the cold temps that would be a problem.
It sounds so easy...
