Ok, for stability, it would have at least 3 or 4 bases rather than two but…
We currently do not have material strong enough to build an orbital tower (aka beanstalk), but suppose we set ourselves a lesser target: getting to say 100km. Do we have material strong enough to build an arch-like structure that tall? Would it need to be a dome?
I can’t imagine any structure made of any material we now can work with being capable of more than a mile or two in height. The vertical stresses and wind shears would be too great even if the supports could hold it up.
What would hold up the dome or arch until it was complete in the first place?
We do currently have a material strong enough to make a beanstalk (AKA space elevator, AKA skyhook), we just don’t have enough of it. Not that this is a trivial problem: The longest nanofiber strands we’ve been able to make so far are somewhere around 10 cm. We do not, however, have (in any quantities) a material suitible for a structure shorter than that but which still extends into space. The problem is that it takes a different kind of strength: For something that goes out past geostationary orbit, you need tensile strength, but for something shorter, you need compressive strength. Nanofiber has plenty of the former, but not so much of the latter. Or in other words, you can’t push a rope.
Building a space elevator doesn’t involve taller and taller towers, though. It’s more like putting a good-sized satellite in geosync orbit and carefully lowering a cable to earth, like a blimp’s anchor line.
Building up to a mere 100km (or anything less than the ~35,700km needed for geosynchronous orbit) would actually be much harder.