My research says crew sizes were at least a hundred men, often two, for a large ship such as a Spanish galleon. Why was this? What is the absolute minimum size a crew could be to get a galleon running? What did the crew do when they weren't at their stations, with a hundred and something men aboard?
For context, what time period are you referring to? It might make a difference.
But, my initial thoughts are:
Hard to give a more decisive answer without more information.
For sailing the ship, the number of crew needed is largely determined by the number of masts and the type of rig. (Also, the details of the rig had some bearing as, generally, later in history, advances in rig efficiency such as divided tops’ls and to’gallants, brace winches, etc., allowed crew efficiency to increase).
Square rig required more crew per mast than fore and aft rig.
Chapman reckons that a 4 masted full rigged merchant ship (late in the age of sail) required a crew of 48. (p. xvi) He reports that the 7 masted schooner “Thomas W. Lawson” (5,200 tons gross) operated with a crew of 16 men.
“Chapman, Great Sailing Ships of the World”, https://books.google.no/books?id=QgMRudqoLGQC&pg=PR15&lpg=PR15&dq=size+of+crew+on+sailing+ships&source=bl&ots=9TyCVTDP9f&sig=E-s9V33IaacETh-vr8BoM_9cJno&hl=en&sa=X&ei=5DwJVcrrDsf4OraggeAG&redir_esc=y#v=onepage&q=size%20of%20crew%20on%20sailing%20ships&f=false
I have sailed a 100 foot brigantine (square rigged on the foremast, fore and aft rigged on the main mast) perfectly well with a captain and 4 sailors. (Of course some evolutions, like setting and taking sail would have been faster with a larger crew, and others such as weighing anchor (man-powered windlass on this ship), would have been easier.)
This was day sailing. On a longer voyage, where 24 hour coverage was required, the same ship, on a 2 watch system, would have worked well with an 11 man crew (Captain, mate, 8 seamen, 1 cook).
Warships, of course, needed far larger crews than merchant ships. This was primarily due to the number of men needed to operate the cannon. A training manual for the French revolutionary navy specifies that a 36pdr cannon should have a crew of 15 men, an 18pdr cannon should have a crew of 11 men, and an 8pdr cannon should have a crew of 7 men.
Chris Henry, “Napoleonic Naval Armaments 1792-1815”, https://books.google.no/books?id=Bs2MVggWyvQC&pg=PT18&dq=number+of+crew+to+man+a+naval+cannon&hl=en&sa=X&ei=UkQJVZObOoWuPKfUgNAC&ved=0CDIQ6AEwAg#v=onepage&q=number%20of%20crew%20to%20man%20a%20naval%20cannon&f=false
Warships only rarely carried enough men to man all their cannon at once (usually only one broadside).
Besides the number of men needed to operate cannon, warships also carried marines, specialist petty officers, larger numbers of able seamen, so that sail handling could be carried out more rapidly, which would be an advantage in combat.
So, the main reason why ‘Spanish Galleons’ needed such large crews was because they were either warships, or armed to protect their cargoes from attack by pirates or national enemies, and they needed to carry enough crew to man their cannon.
The absolute minimum size of crew to get a galleon (assumed to be 3 masted, full rigged) running I estimate at 5-6, (but that would be very slow and inefficient – Setting or taking sails would be one at a time for example – This might not even be enough manpower to haul the anchor (depending on windlass efficiency and strength of wind and wave) but the cable could always be cut in a real emergency.). For fairly efficient operation (one watch only – not very good for long voyages) perhaps 12. For fairly efficient sailing on a long voyage (but very inefficient fighting with the cannon) about 25.
(Note: "Fairly efficient" means "a reasonable balance between the cost of the crew and the efficiency, performance, and safety of the ship", as it might be perceived by a merchant ship owner (perhaps a somewhat stingy one). Clipper ships, (also three masted, square rigged ships, like galleons (though 200 years or so later in history)) on the California run, generally carried a crew of 50-60. These ships carried proportionally much more sail area than Galleons. They valued speed much more highly. They knew they would need to sail through the extreme and dangerous conditions round Cape Horn (where a larger crew which could take sails rapidly might contribute greatly to the safety of the ship). So they were willing to invest in a larger crew in order to gain "high efficiency" rather than "fair efficiency". Different ship owners in all eras might value sailing efficiency differently. The more they valued this, the more they would be inclined to ship a larger crew.)
(Rough estimate for number of crew required to provide various "efficiency levels" for a three masted, full rigged, ship on a two watch system: "Fair efficiency" - 25-30 men; "Good efficiency" - 30-40 men; "High efficiency" - 40-60 men.)
For what did the crew do when they weren’t at their stations, see this earlier post which asked a similar question but about 1st and 2nd rate ships rather than galleons. The tasks I mentioned in my reply would have been very similar for galleons.