No need for the saucer to accommodate "all" of the stuff, as the ship also has the sizeable engine booms for the purpose.
Most of Kirk's ship is empty space, literally - space outside the hull accounts for most of the volume within the box that the greater length and height indicate. Kirk's ship also has bigger engines, main culprits for the greater length, but people aren't berthed there in the general case. In terms of supposedly habitable volume, the difference is minimal... Certainly much smaller than the difference between 430 crew and 80 crew.
From the book Federation: The First 150 Years.
I'm sort of disappointed they went with the Famous Scientists theme - a theme of Rivers of Scotland would have been more interesting.