You have to have a rope several hundred miles long that won't break. That requires a bit more strength that your six ply tow rope from Home Depot. Even if, as the tow rope advertises, it can absorb Towing Jerks.
Our good industrial grade materials currently top out around 20 GPa (giga-pascals, no don't ask me to describe a giga-pascal), but the space elevator needs a bit more. The best guess is 62 GPa. No we can't use three times as much and make up the difference, cause three times as much rope is just three times heavier. Luckily carbon nano-tubes are really strong, like 220 GPa. "So lets get started," you say. Well, not so fast. So far the record length for a carbon nano-tube is about four centimeter. Its not quite long enough. But they're working on it.
In the mean time, its not all snoozes over at the material science lab at NASA. Not at all. As New Scientist reports, even if you have to take a rocket up to low earth orbit, you could take advantage of a slightly shorter tether to whip up into higher altitudes without any additional fuel.
Imagine a satellite (with considerable mass), solar panels drawing power from the sun, and a large loop of wire that sweeps through the earth's magnetic field. The solar panels generate electricity which is used to create a counter magnetic field in the loop (like a large motor), accelerating the satellite forward little by little. Over time this would add up to a tremendous amount of kinetic energy.
Now, at the right time, the satellite drops the loop (or another wire), down into the lower atmosphere, and just as its starting to drag; it picks up a passenger off of a low earth orbit launch vehicle. Could be a capsule with a person even. At that moment the capsule or package starts dragging on the satellite, slowing the satellite down, but speeding up the capsule. The timing is a little tricky, and the satellite has to be large enough that this step doesn't drag it out of the sky, but when everything is in balance, this can be used to put the payload into a higher orbit, or even escape the earth's gravity well altogether.
They've already done some experiments to try this out. It does work. Ok, so on STS-75 the tether broke, but then again the suspended probe was generating 3500V and that can make quite a spark.
This next test is going to use something stronger that doesn't conduct. No, that's not dental floss, its fishing line.