If you had 250 miles of wire (only about 400 pounds), could you unroll it all the way down to the surface of earth and drag it across the world? I guess at some point it would melt from friction with air. But what if the source was geostationary??
No. Any type of metal wire would break under its own weight.
What you're describing is essentially a space elevator. Interestingly, such a system wouldn't actually end at geostationary orbit - its center of mass needs to be in geostationary orbit, so the top end would need to be significantly higher up. It might be possible to construct such a structure using exotic materials such as carbon nanotubes, but even that would have to be tapered to achieve the needed tensile strength. Ordinary materials like steel are out of the question.
If you want true insanity there's a way it could be done.
As you say, the span is too great for anything to be strong enough to hold itself up without the taper becoming completely impossible. You have to break it up into much smaller chunks--impossible, you say? No. Build a ring around the Earth at the equator inside an evacuated tunnel. Spin the ring at a speed sufficient to generate an outward force (maglev setup to transfer the force) sufficient to put supports in tension. Do the same thing again a bit farther out. Again and again. I haven't done the math on it but as the strength requirement goes to zero as the towers/rings go to infinity it's simply a matter of building enough of them. Yes, orbiting rings are unstable, but this is tethered.
And you can also build the Ringworld that way--no super materials, just an extremely massive stationary track underneath to provide the support. But, not being tethered it has the same instability problem. You can't make the walls but you can slope it without undue forces.
