Even assuming that it were physically possible, a hurricane would test this concept dearly. Even a small surface extended over such a distance would react violently to wind, especially if it were ribbon shaped as it is often illustrated. Additionally, hundreds of kilometers of anything will have a substantial weight, not to mention the weight of the elevator. Any weight must be overcome by the pull from the space station, so it has to be in an orbit that is unbalanced with respect to it's orbital velocity and distance. There's no reason to expect it to stay over the same point on earth, which is the assumption made when the idea is imagined. What if it were tending to travel faster than the rotation of the earth in order to create the force required... Wouldn't the tether then be at some angle other than the zenith, as it held the space station back? This concept seems fatally flawed.

I don't know it is necessarily flawed. It was advocated by Arthur C Clarke (who figured out geosynchronous communications satellites in 1945) and by Robert Anson Heinlein, another SF writer who was also a "proper" engineer.

Anything in geostationary orbit has to be 36,000 kilometres up, and directly above the equator. Any lower, and it has to orbit faster than a 24-hour day or it falls down. Any higher, and it will lag the Earth if it is in free orbit.

However, a body above 40,000 km and tethered should stay pretty much vertically above its tether point due to centripetal force. The extent of that force can be tuned by the amount above 36,000 km, so it should be possible to find a height that would minimise instability.

For stability, fail-safe and maintenance, multiple tethers would be needed, which would allow some aerodynamic configurations to be used. The usual idea is to have a set of 4 tethers to an elevator. Power for the elevator to wind itself up and down is provided through the tethers, using supply from the ground for the first few km, and then having automatic attendance by solar panel units above the atmosphere. You would want elevators in pairs to share power by regenerative braking.

The present limit is the strength of the tether. The best steel rope for suspension bridges can only carry about 20km of its own weight. That is, if you unroll a rope from the top, when there is 20km hanging, it will snap off at the top. Making it thicker makes it heavier in the same proportion.

However, steel breaks because the material cannot be uniformly made. There are crystalline fractures. Monofilaments - single molecules of chains of some elements - can theoretically be much stronger. The theoretical limit to the strength of "multiwalled carbon nanotubes" is about 210 times that of pre-stressed steel wire strand, and it is about 6 times lighter. So it should be posible to make a self-supporting tether about 26,000 km long with current-generation technology. Getting another factor of 2 should not be an insuperable problem. Even tapering the tether top-to-bottom might make a significant difference. So a space elevator may be practically feasible.