Ya the space elevator idea has been around for multiple decades.
With carbon nanotubes it could potentially be possible. However building a long enough cable is a problem since it's not an easy material to work with.
I don't think the cost would be much of a problem honestly. The ISS + shuttle program are/were pretty insane and it still got done. The investment risk factor is just pretty high.
Also think about how much cheaper computer parts or superconducting wire is compared to 10 years ago. If the demand is there automatation will dramatically decrease the price.
Though another thing would be the effects such a construct would have on earth itself. Even a small (in relation to earths mass) counterweight could imbalance earths rotational speed and posssibly orbit.
I'm sure there are decent simulations about that but in the end you might have to abandon the project if the effect was greater than expected (butterfly effect so to speak). It's unlikely but in the end our understanding of the core of the earth is only a simulation without proof.
Anyway if the human race hasn't killed itself off in a hundread years I think this could potentially be built. In our lifetime? Probably not.