Your length is off by three orders of magnitude. Fortunately, the surface area is off by four. In the most serious proposals, the space elevator "cable" is a flat tape with a cross-sectional area of a few square centimeters. Thus, we get:

2 cm^2 * 10^8 m = 20000 m^3. Yes, it's a lot of nanotubes, but the price of nanotubes per unit mass has dropped exponentially for a long while now, and the economies of scale involved in building a space elevator will lower the prices even more.

It will likely be much thinner than that. But that is a problem. Most plans assume "once manufacture becomes possible and cheap enough". Of course, ordering that much (once a process is proven) will quickly drive down the price. Talk about scale!