One day far far in the future, and probably on the moon first. The resources and technology needed to do that are far beyond what we could muster at our current rate of space tech development. It would take like an asteroid discovered 125 years away from impact to mobilize the entire globe to work on the thing and actually get one working, if it would be possible.
Also, for anyone that is confused, a Space Elevator is a sci-fi idea that has some facts based in actual theory. A space elevator would be pretty much like a giant cable attached to earth's equator that reaches out all the way into space, and is attached to some kind of orbiting counterweight. The tension on the thing would be incredible, beyond what we can do right now, and the thing would actually have to be literally 'dropped' out of space and 'landed' on the ground, another technological nightmere. It would make getting something into space very cheap and energy efficent however, and would be essential for any kind of long term human space endevor.