Edit: VLM does have a point in that a falling cable might be a problem on Mars. The moon doesn't have enough rotation for a fixed skyhook (space elevator) anyway. :-) More advanced rotating elevators in a moon orbit, maybe. But it might be hard because of the moon mascons (the moon is lumpy internally with different density, which results in gravitational anomalies).
The moon doesn't have enough rotation for a fixed skyhook (space elevator) anyway.
You're just fairly restricted in where you can put it- you can only put one near the sub-Earth and anti-Earth points, such that the cable passes through the L1 or L2 Lagrange points, rather than anywhere along the equator. The cable would have to much, much longer than for a space elevator on Earth, but the Moon's lower gravity actually makes the engineering challenges Not As Bad- i.e., we could build a lunar space elevator with current materials.
On the moon, an electric rail launcher with 5-10G only needs to be a few km long.
(Iirc, 1G acceleration of four minutes is the launch velocity of the moon? ~ 2.4 km/s.)
Edit: This assumes local materials for the electric launcher. If everything is skipped from Earth, it might be different. But, consider -- if you need to launch so much stuff from the moon, you must already have the infrastructure to build things there.