Feasible Space Elevators
Trying to span the depth of the Earth's gravity well with a single structure is terribly non-optimal from an engineering standpoint for a number of reasons. Smaller elevators, however, are quite feasible with off the shelf carbon fiber.
Imagine a rotating cable with a tip velocity of 2400 m/s, and a comfortable 1 gravity at the tip. The radius then works out to 587 km. You don't want the tip to enter the atmosphere, so you set the orbit of the center to 750 km, and the lowest point is then 163 km. The center then orbits at 7.48 km/s, and the tip is moving 2.4 km/s less or 5.08 km/s. Subract the Earth's rotation and you have a velocity of 4.61 km/s relative to the ground.
Your launch vehicle now has a much less challenging job than getting all the way to orbit by itself. It merely needs to reach a landing platform at the tip of the cable. To return to Earth, it has much less velocity to dissipate, so the re-entry heating is much less. Cargo heading to higher orbit merely rides for half a rotation, then lets go. It now is moving at 2.4 km/s above orbit velocity, which puts you in a high transfer orbit.
The load on the cable varies linearly from center to tips, therefore is equal to a cable under 1 g half the length, or 293 km. Good carbon fiber has a breaking length at 1 gravity of 360 km. We want a decent margin of safety, so only load the cable to 40% of ultimate strength. This requires tapering the cable from center to tip, as each point has to support the payload + cable outside that radius, but the taper ratio is not severe, about 7:1 in area.
Because this design is 40 times shorter, it is much less exposed to meteor and debris damage, but they still are a risk. Therefore you build the cable out of something like 21 strands, of which 7 are spares, and cross-connect them every 5 km. So when the inevitable impact happens, you only have to replace the one 5 km segment, which is 0.02% of the total structure. The "single cable" illustrations like the one in this article are just terribly unrealistic from a safety standpoint.
This type of rotating space elevator is called a Rotovator, and can start being useful while under construction, so you don't have to build it all at once. As the length increases, and tip velocity goes up, the launch vehicle needs less velocity, and can therefore carry more payload. If some of the payload is more cable for the Rotovator, the increased cargo on later flights "pays back" the payload spent on launching the cable. This payback time in cargo mass can be relatively short.
You can build a second Rotovator in high orbit, which captures payload sent up by the first one in low orbit, and forwards them to interplanetary trajectories. Since kinetic energy is not free, you need onboard electric thrusters to maintain the Rotovator orbits, but those have ten times higher efficiency than conventional rockets, so you still come out very much ahead on net cargo.
