Depends on the threat situation and the weather (which can be a threat situation!) and of course the trade/position. I'll only speak to at sea.
Weather is bad, and everyone is seasick there will be a "pipe down" where if you are not on watch or essential pers go to your rack and try to not vomit. Minimum crewing levels.
When the threat situation is low the watch rotation will be minimized so that more people will be off watch at a time than on watch. Get some PT in, or relax a bit etc... I've seen the Ops room basically empty with people on call instead of just sitting at their station.
When the threat situation is high you go to what we call 1 in 2. (7hrs on , 7 off, 5 on , 5 off) Which basically is half the crew awake and at stations at a time. All the weapons and sensors are crewed and ready to fight. If we go to emergency or action stations the rest of the crew jumps too and goes to their assigned station. 1in 2 can only be done for relatively short stints (like a week) as it can get exhausting.
There are new watch rotations now like crazy 8's and 4Alpha which all have different awake and off watch times.
The non-Ops types like Naval Technical Officers, Dept leadership, or the Logistics Dept (all ranks), are called day workers and run a normal daily routine. Of course at sea you are often working late or on call, particularly the NTO's who can't just sleep through an engineering problem (unless the on-watch PO1 has it covered).