Response from Apoc at 11:30 today:
The servers are back up for now, and we've finally tracked down the root of the issue.
Non-technical answer: There is a bug in a 3rd party library we use that causes all SQL connections to randomly stop, which causes our auth servers to be unable to "sync" with eachother, and retrieve data when required. This causes them to basically stop responding due to failsafes to avoid bigger issues (such as melting server hardware).
Technical answer:
We use MySQL's .NET Connector to handle our SQL connection (and queries) on the auth servers. There is currently a bug in the entity framework support that will cause connections to stop working after what seems like a set amount of time, or number of requests to the DB server. We've identified the issue, and are going to make changes today and tomorrow to remove the entity framework from our server software. This should resolve the problem for good, at the cost of some maintenance annoyances. We're not sure of what exactly is the cause in the MySQL connector code-base, but we do know that is where the issue originates. Unfortunately, this is one of those times where relying on a 3rd party implementation (that has been tested to work perfectly) can cause a nightmare. We must be hitting some usage case that hasn't been tested, or found a new bug in the provider.
TL;DR; We found the problem, and we'll be fixing it ASAP. Servers are up for right now, but may go back down again later today. Please bear with us!