This is slighly less mysterious if you know that the zero-point of unix timestamps (which is how Moodle stores dates and times) is midnight 1st January 1970 UTC. Now, express that in your time zone, and presumably you get some time in 31st December 1969.
So, what is acutually happening is that Moodle is failing to save the date, for some reason, and then treating that missing date as 0 when it tries to display it.
I am afraid that does not really answer the question about what is going on, but it may explain a bit.
Now you have to work out what action it was that failed to register the right date. Can you reconstruct what was done? If you repeat those steps with a test user account, does the same thing happen?
A bug that can be reproduced is much easier to fix.