Check out this illustration: that's a long way down! 442x440, 2048x2048.
It sounds like some of the front-runner tether systems don't actually attach to the ground: if you travel the last few dozen miles with an aircraft and dock, then the tether can be shorter, and doesn't need to be in a geostationary orbit. This means the tether actually orbits the globe, so launches can rendezvous with it whenever it passes overheard, and it also needn't be at the equator.
The weirdest variation is the "rotorvator" model, where the tether spins like a bicycle wheel, picking things up at one end and depositing them at the other.
I think one of the coolest things about these tether systems is how they interact with the Earth's magnetic field: if the cable is conductive, then its motion through orbit will induce a current, just like how generators work, but unwrapped. However, if I understand it right, the field associated with the current causes drag by its magnetic interaction with the Earth's field. And drag means you slow down, meaning your orbit decreases, so it's not a good way to generate power (you trade orbital elevation for electricity.) However, if you inject electricity into a tether, then you end up pushing against the Earth's magnetic field, and turn magnetism into lift: so it's a way cheaper way to push things into orbit than chemical rockets, especially if you're not in a hurry, and can rely on solar power to do it.
The space.com article claims that one could be built for $10B. Which sounds like a lot, except when you consider that each Shuttle costs something like $2.5B, and each launch costs $17M.
You know, when I was 14, I didn't even entertain the idea that it might still be impossible for me to be living off-planet by the time I'd reached my current advanced age. What the hell went wrong? Won't someone drop a cinderblock on the gas pedal and get me to the future already?