It's an engineering problem - and engineers may have solved part of it.
One of the biggest obstacles to building a space elevator is finding a material strong enough and light enough to form the tether (It's not called a cable - when something holds two objects in space together it's a tether, even if one of them is a planet).
They were experimenting with benzene - and apparently if you compress benzene far enough then release the pressure slowly, it can turn into a chain of single carbon tetrahedrons. Otherwise known as diamonds.
A literal string of diamonds hooked together. Right now, they can only make very short strands, but if they can scale it up then this could be the answer.
So, maybe somebody should nickname the first climber "Lucy."