The root partition claimed 100% utilization, which prevented exim from writing /var/spool, and prevented emails from queuing. This caused exim to return a 4xx error, which is good and what it should have done to ensure that no email is lost. The issue will have delayed inbound email, but it will eventually be received so long as the sending servers function according to spec, and there is no reason to expect otherwise.
However, the root partition was never at 100% utilization despite reporting that. Currently, I have no explanation for that, but will continue investigating.