It is already possible to build a space elevator. The key idea is the Orbital Ring version of the space elevator, not the geosynchronous tether concept you are familiar with.

See, for example, Paul Birch's writings:

http://www.orionsarm.com/fm_store/OrbitalRings-I.pdf

There are many other feasible ways for us to get to space much more cheaply and safely than rockets:

https://en.wikipedia.org/wiki/Launch_loop

https://www.youtube.com/watch?v=J1MAg0UAAHg

https://en.wikipedia.org/wiki/Mass_driver

https://www.youtube.com/watch?v=KerG4ILWEa4

https://en.wikipedia.org/wiki/Space_fountain

https://en.wikipedia.org/wiki/Space_gun

You get the idea. Any of these could be done for $100 billion or less, a negligible cost when considering a GDP of trillions of dollars.

Why do something like this? Overnight, asteroid mining becomes an incredibly profitable industry. Because the cost of delivering payloads to LEO drops from thousands of dollars per kilogram to less than $100 (potentially less than $1) we can now retrieve asteroids with trillions of dollars worth of minerals for mere tens millions of dollars in addition to having an easy viable way of returning those resources back to the surface. We acquire the ability to deploy profitable solar power in orbit above cloud cover and with the ability to return said power back to the surface with near zero loss by running power transmission cables down the elevator.