I still need to run one last test with another phone to make sure that it's not the phone jack that's suddenly died for some unknown reason. If it has, that's going to be a major pain. All in all, I'd rather buy a new phone / answering machine.
Update: And the answer is: it's the jack that's dead. The answering machine has been moved to a working jack and there's going to have to be some wiring investigations after Capricon. Ah, well.