People got enthused about this after Brad Edwards, mentioned in the article, led a NASA study on space elevators about a decade ago, which found it to be surprisingly feasible. Here's their final report (pdf): http://www.niac.usra.edu/files/studies/final_report/521Edwar...

The material would be carbon nanotubes a couple centimeters long, bound together by a reasonably strong epoxy. They estimate a cost in the low tens of billions of dollars, by using seven shuttle flights to deploy a minimal elevator, and using that to pull up additional elevator material. They examine a lot of practical issues and find solutions for them.