Currently doing a Very Hard run and I can't save at all, even to quit the game. Which really sucks, as I was just about to log off and wanted to save all the stuff I'd done since the last run.

The button to save is greyed out, and so is Save & Quit. Can load games just fine as far as I can tell. Haven't had any major bugs, so no idea what caused this.

Usually saving is only disabled during conversations, the enemy turn of combat, or similar situations where some longer scripts are executed.
So ending the dialog or if you are in combat waiting for the start of your turn should allow you to save.