Error 503 is returned by plex media server after it gets into severe overload with thousands of queued requests. This can arise after a deadlock - where one thread may lock out all other threads
To help identify the cause and start an investigation, we would need logs + process dump + list of connections at the time.
To start ensure debug logging enabled and also crash reporting enabled. Restart the server to get fresh logs and follow the instructions in this post (one of several with deadlock diagnostics info) to gather diagnostics for the suspected deadlock
