Is there a website which literally adds up the bits within the Third stage that shows it does all fit
You could go to the source:
Apollo Saturn V Flight Manual.
Although this site works as well, with some other historical context. All of the following diagrams are taken from there.
The
Saturn V rocket is everything from the Instrument Unit on down. The LM is not part of that volume.
These are the rocket stages, with dimensions:
S-IC (first stage):
S-II (second stage):
S-IVB (third stage):
S-IVB tank construction:
As is clear from the diagrams, engines and pumps took up space in the interstages; some lines were routed through the tanks, some around the tanks, etc. Other diagrams at the site show how tanks were constructed, how engines were mounted to thrust structures, how fuel flow worked, etc.
It's not light reading (unless you're a freak), but it shows that yes, Virginia, you could fit everything you needed within the Saturn V's volume (remembering that the spacecraft
are not part of that volume).
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