NASA did a study on space elevators about a decade ago, and found that it would require carbon nanotubes several centimeters long, bound together by a realistically strong epoxy. Launching their design would require seven space-shuttle flights to deploy a minimal elevator, which you use to pull up additional construction material.
They addressed a lot of other practical issues too. Here's their final report (pdf), it's an interesting read.
Ah neat, that looks like an extension of the book he wrote.
I get a bit worried when he dismisses chemical rockets for lifting the initial spool into space, and starts going on about electric drive... it seems like he should limit the number of technical revolutions required to make this work, if he wants this to succeed on a reasonable time scale...
They addressed a lot of other practical issues too. Here's their final report (pdf), it's an interesting read.
http://www.niac.usra.edu/files/studies/final_report/521Edwar...