Just kidding
It turns out that I had 2 problems at once, and I don't know which caused the other. At around 9am, Sprint (our bandwidth provider) stopped responding, and it took several hours to get it fixed on their end. But then once it did, it looked like the hard drive on the server was corrupted!
I was able to fix the error on the hard drive, only to find that it had caused errors in the database.
I think that I have all of those errors fixed now. So unless something else pops up, I think that we're back to normal.