If the cable were 1/4 inch thick, the weight of it from earth’s surface to non-gravity orbit would be hundreds of tons, and it would break.
And how long would it take to travel a nquarter million miles on a cable?
Such a cable would have considerable mass. So to stop it from falling, it would have to be balanced at the other end by a similar orbiting mass. The entire elevator would then be supported by centrifugal forces.