Would a beanstalk (or "skyhook") be possible on the moon?

Yes, but it would have to be extremely long. There is no simple "lunar synchronous orbit" that is analogous to a geosynchronous orbit, since the moon rotates only once per (lunar) month, and the near side always faces the Earth. A lunar beanstalk would go to the nearest lagrange point, which is Earth-Moon L1: between the Earth and the moon. (This can also be called the cis-lunar lagrange point.)

The L1 Lagrange point itself is mildly unstable, but of course a beanstalk would be longer than the distance to the Lagrange point itself, and would extend into the Earth's gravity well slightly to keep it in tension. The beanstalk would be terrifically long, a minimum of 58,000 km just to get to the Lagrange point, and then depending on the counterweight mass, a bit further. However, because of the low lunar gravity, a Lunar beanstalk doesn't require an impossibly strong material to manufacture.

Information on Lagrange points can be found in the gravity reference pages at: http://www.merlyn.demon.co.uk/gravity.htm#L15.