After some research, combing of the logs on the server, and reading of documentation, we believe we found a reason that the web server stopped working, but we believe the cause evaporated when, in our desire to get everything working again as quickly as possible, we just rebooted the machine after other options to restart the web server failed. However, even the suspected cause does not fully jibe with the observed symptoms.
The good news is that we have recently commissioned a new server (NC030) to take on the websites hosted on NC020. We are still working on setting up the control panel on that, but we hope to have it in production as soon as possible.
We apologise for this incident, and thank you for your patience and understanding.