Keep in mind that we only see officer's quarters in TOS, and don't see the crew quarters (which are six-bunked and more like a college dorm's) until TUC. Yes, the carrier Enterprise is cramped and tight, but if you remove 4/5th of the crew and get rid of the fighter hangars and flight deck (but leave the helipad and garages), suddenly it gets a lot more spacious.
The enlisted quarters shown in TUC are more spacious than officers' quarters aboard CVN-65.
This is a junior officers' compartment aboard the Midway, absent the additional lockers that would be placed against the bulkhead. On the Enterprise and later supercarriers, these sleep three officers, rather than two.
This is a senior officers' compartment aboard the same ship, seen through a cutaway made for ease of museum viewing. Compare this two-person compartment with Uhura's multi-compartment stateroom (seen in "Elaan of Troyius"). Though she would be classified as a junior officer by modern standards, her quarters are several times the size of those given to two senior officers aboard a modern carrier.
Removing the hangar deck from CVN-65 would noticeably reduce its volume, but less than you might think. Keep in mind, too, that the TOS
Enterprise has an internal shuttlebay, shuttle maintenance spaces, and vast, essentially uninhabited nacelles - which CVN-65 does not. The flight deck, despite being the center of carrier operations, is not counted toward its volume, since it's external to the ship's structure. (There is no helipad, by the way. Helicopters operate from the flight deck.)
Space is simply used wastefully aboard the TOS
Enterprise, to an extent that its implausible for a ship of its size (that contains everything shown or implied - e.g. quarters for 430 crew - onscreen).