Well, 'balance point' would be at geosynch, which is like 42k km up there. A base station would be sunk deep into the bedrock, maybe 50 floors deep with anchor pits going perhaps a half kilometer. At the top, balance mass would be strung out beyond the balance point so the whole mass would be effectively in orbit. Its center of mass would not be at geosynch once finished, though, due to some rather interesting orbital dynamics. To keep a bit of tension on the cable, mass (another station) would be put at the "top", to tighten it after the fashion of a rock spun on the end of a rope.
The mass has been defined, approximately; it would be a minimum of a million metric tons of just cable; the rest would depend on the structures hooked to it.
BTW, there's already a consortium of businessmen with funds pledged to build the thing once a material is available to make the cable. Labs all around the world are working on "buckytubes" of carbon, which are calculated to have more than enough strength. If built, the cost to get a load from the ground to geosynch orbit would drop from the order of $11k US to about $250 US.
Whoever builds it could effectively own the solar system, since cables could then be dropped on Mars, etc. -- and loads to the outer system could be launched by the simple expedient of dropping them off the outer end.
Heh -- I just edited the Wiki article....
note: building such a thing would employ a LOT of people!