This, and the various other problems (iTunes and other apps would not start), was caused by a very badly incorrect date on the system.
Somehow the date had been set to 27/08/12010, so that’s about ten thousand years in the future. Some programs didn’t like that, understandably, as it’s more digits than I’m sure they’re ready to accept.
In the attached image you can see how the Windows Event Log interprets this date. Apparently 2036 is about as far in the future as it can go.