Talking to a Merchant Navy friend earlier and it looks as if Enterprise might be shorthanded!
If we just take the Bridge and Engine Room, there should always be a senior watch officer in command. Thus no deep blue vessel can sail without at least a Third Mate and Third Engineer, all of them have to be certified by maritime law (Passenger liners go down to a fifth and sixth, but the cargo has a habit of moving around so requires more care).
The only time you will find them on the bridge together is when the ship is entering or leaving harbour, or when it is sinking.
All this excludes the Captain, who does not have a standing watch at all. Captain's have very little to do with the running of a ship. Third Mate handles cargo, Second crew, First the ship.
Then you need a bridge crew. On Enterprise this appears to be a minimum of 3: Driver, Comms and Lookout (or whatever the T'Pot is supposed to be). If we assume standard 8 hour watches that gives us 12 crew to man the bridge. Plus the Captain.
In Engineering there should also be a full operational crew, plus the shift engineer. There seems to be a lot of people in Enterprise's engineering section, so we will say half dozen on the watch. That is 21.
Enterprise has a combat role. So it requires the plebs to handle that. Again it has to be trebled up. There should always be a crew in the magazines and weapons platforms. Say two for each torpedo and two for the ion cannon stations. Plus Senior Weapons Officer to control them and somebody to handle fire control. 30
Enterprise appears to have a galley. At least 2 for each shift, plus the Captain's own chef and steward. Say 9 in total.
That gives us 72.
On top of that there is the general ship upkeep, damage control, science and provost branches, the sawbones.