After skimming over this thread, I did not see any mention of Arthur C. Clarke's "The Fountains Of Paradise", which provides an excellent treatment of the construction of a space elevator. Since few people here are likely to read it, I will try to provide a synopsis.
Development of an ultra-strong synthetic material (Clarke used diamond monofilament) is made possible by zero-gravity manufacturing. Starting from geosynchronous orbit (which I call the Clarke orbit, because it is easier to spell,) two tapes, or cables, are constructed, one going up, the other going down. The mass of the structure must be kept in equilibrium to prevent it from wandering off station. A counterweight of several million tons is attached to the upper end as the lower end is anchored on Earth. The structure is a square tube, with tracks running up each side. Payloads are not lifted, they propel themselves up the tower, or are allowed to descend with braking forces. (Regenerative braking is used to capture energy.) Payload costs are equivalent to running an electrical motor of a few horsepower for about 30 hours, because the vehicle is not having to hold itself up against gravity, nor does it have to deal with atmospheric resistance. Payloads for other planets can be launched by the tower acting as a sling.
Breakage of the tower does not result in the whole thing falling down, because the upper portion is kept in orbit by its velocity. Only the lower few miles of the tower is subject to atmospheric disturbances, which would be the most likely cause of a failure. The tower would be a flexible structure, which would result in bowing, or distortion, as payloads ascend and descend. The capability of launching payloads by sling is limited because of the energy transfers involved, which cannot exceed certain values.
Clarke took a huge leap of speculation in '3001', supposing that it would be possible to connect towers together at the Clarke orbit altitude, eventually forming a ring around the planet. The towers are still supported by the orbital structure, and are not held up from below.
Manufacture of the material used to support the structure is almost certainly going to have to happen in space, because thousands of miles of CONTINUOUS cable is required, and the cable must have a tensile strength of thousands of tons per inch. The amount of carbon required, as well as the energy needed to process it, makes it unlikely to happen on Earth, even if the problems of gravity effects on the manufacture of the cable could be overcome.
Once constructed, no other method of reaching space could be as cheap as the tower, because any vehicle must hold itself up against gravity while traversing the portion of the atmosphere which is too dense to pass through at high velocity. A vehicle climbing the tower only needs enough energy to climb against gravity, not to hold itself up as well as climb.
Don't expect to see a space elevator in construction until we have cheap and easy access to space. But the amount of commerce which will happen between Earth and space will eventually not only justify constructing an elevator, but necessitate it. Where one or two people used to walk or ride horseback, now thousands travel at a time. The same will be true with space. Where one or two traveled by rocket, someday thousands will ride by elevator.