A cable from the surface of the Earth to a satellite. I’m sure the engineers have calculated the stress that would be on the cable due to gravity, and the weight of the cable itself, plus the effect of winds and the movement of the cable through the atmosphere because it is being dragged by the rotating Earth to which it is attached. Also, they will have carefully considered all possible scenarios in which something can go wrong and have assured themselves that the humans in the space station are not in danger when the cable snaps or a plane flies into it or a hurricane or a tornado whips it around or lightning hits it.
Engineers think of everything, and they never create structures that collapse.
One Doctor in a Bonanza and you’re screwed.
It does not necessarily need to go all the way down to the surface of the earth in order to be useful.
Lets say we have a platform, just above the atmosphere so we don't have to worry about air friction. The platform is connected to a satellite in a much higher orbit, and as a result it is traveling at a much lower speed than it would need to, in order to stay in orbit.
Being able to get a spacecraft to that platform without having to achieve full orbital velocity would save a lot of fuel.