What’s to keep the whole thing from flying out into space or crashing down to Earth?

The upper half of the elevator gets thrown outward [by Earth’s rotation], and the bottom end is pulled down by gravity. Even if you cut it at the bottom, the ribbon would just float there.

How would the elevator work?

A climber [vehicle] would have two treads to pull itself up. It would travel at 193kph. So if you want to go 35,000km up, where the telecommunications satellites are, it would take about eight days. If you want to go to Mars, you would go all the way to the end of the ribbon [100,000km up], let go at the right time, and it would throw you to Mars at a high velocity–you wouldn’t need a large rocket to get there.

How can a paper-thin cable support so much weight?

It requires carbon nanotubes, which are 60 times stronger than steel. [Some researchers] are working on putting nanotubes into a composite material within the next couple of years.

How soon could this be built?

I think 15 years is a realistic number.