My guess is that we used up all our CPU credits, which causes the site to slow down drastically. This, in turn, triggers an automatic restart of the forum process. Unfortunately, restarting just uses up even more CPU credits, making the problem worse. But this time, the server was far enough gone that I couldn't verify the problem, and I just ended up replacing it with a new server.
The long-term fix here is probably to either dig into the admin panel and ban some badly-written search engine crawlers, or to upgrade to an even bigger server with a permanent CPU allocation instead of CPU credits.
My apologies for how long it took me to get the forum back online.
