The Apache web server and all websites were for about 30 minutes.
The server was unable to handle very high load, and Apache could not recover. This follows some new Apache configuration settings on our side to better protect against website attacks. Ironically, it would seem it is exactly those attacks (20+ concurrent WordPress login attempts at any moment) that caused the high load and brought the system to its knees.
Apache is back online with its previous (stable) configuration. We will do more homework and come back later this week with fresh tweaks.