https://www.youtube.com/watch?v=T02t_MBgYVU
Contest in Japan for cable cor for elevator. These have run fro several years. The Japanese are serious about a space elevator.
Hard part will be parking a sizable asteroid in earth orbit to anchor the top end of the elevator...
I thought that was unnecessary due to the mass of the elevator itself.
For reference, picture an axle spinning with a chain attached to it. It should be able to enter a stable stretched state.