Unfortunately I don't think carbon fibre Is even nearly strong enough.

If you mean a moonthread through the Earth Moon L1 Lagrange point to stabilize a station there, yes, that's doable. Then you can use electromagnetic cannons to one-shot payloads of lunar materials to that point (coasting up to the gravitational saddle point and coming to rest there with precisely the right velocity.)

Was just about to post the same comment!

I presume we have other obstacles to overcome first ;)