I've also had this problem on PS4. Weird things definitely seem to happen when you've travelled, one time I came back and my sim just stood on the pavement with her arms straight out but there was noises and stuff like she was going to the bathroom and cooking when she wasn't moving even though the actions bar showed she was doing these things. Then I went on a date and nothing happened we got to the venue and the guy just totally disappeared. I'm not sure exactly when the save function stopped working but the last autosave was an hour before i stopped playing and that last hour I had travelled a lot.
1. I was in live mode, had just got back from work and was asleep brought up the menu to save clicked on the save file to overwrite and it just bounced me back to the pause menu with no cursor, had to click down the L3 to bring up the controls to get the cursor back. Tried to save on another slot but the same thing happened again, tried to travel to kick in the autosave but that didn't work either.
2. I have 1 sim in my household
3. I bought the Streamlet Single furnished and have added some furniture, I also renamed the lot when I first moved in but it has saved since then.
4. I had travelled a fair bit in the last hour to go to the festivals