Update: So it's a good news, bad news situation.
The good news is that the server is still up, it's just not externally reachable right now. Not a hardware failure or anything like that.
The bad news is that it was intentional. Evidently auto-payment failed due to insufficient funds. The host claims they sent an email the day before they disconnected the network adapter (not much of a warning there...), but it was not received so I wasn't aware of the problem in advance.
They can also be slow to respond to tickets and they do not respond after hours or on weekends unless you pay for their managed support package. Which ADC doesn't have because it's rather expensive and we rarely need them to do anything. That's why it took so long to get to the bottom of things.