This is most likely the result of a deliberate or (in my case) accidental upgrade to WordPress 3.0, with which BuddyPress 1.2.4 is NOT compatible with.

The straightforward answer: downgrade WordPress 3.0 back to 2.9.

1. SSH into the server
2. Download WordPress 2.9
3. Go to the installation directory where WordPress 3.0 is installed.
4. Back everything up
5. rm wp*
6. rm -fr wp-include wp-admin
7. cp -fr wordpress2.9/wp* wordpress2.9/wp-include wordpress2.9/wp-admin .

That should do the trick.