1. Nudge a big ass asteroid into low earth orbit without accidentally destroying all of human civilization.
2. Develop building materials with sufficient tensile strength to prevent the building from breaking loose at very high velocity and destroying anything in its path.
3. ....
why do I think that thing will build up a static charge and cause a spark between it and the earth?
Just put an asteroid at the lagrange point between Earth and the moon with a reactor large enough to create a protective magnetic field(from cosmic rays and solar wind). Build a city there.