Taking some numbers from EU sources, I find a the Battle of Wolf 359 was "forty starships" - and that was over 10% of the fleet. Most of which had smaller crews than the Galaxy class. But even presuming 400 Galaxy Class Cruisers, that would only be 400,000 people. If we presume 1/2 are officers (unrealistically high, but until late in TNG we're not told of enlisted), and that half of those are ensigns... and that Ensigns remain so only 2 years... we're looking at 100K officers, 50K ensigns, and 25K ensigns per year...
We do know, however, that most of the ships are under half the crew size, most of the ensigns serve as such for 3-6 years (much slower than in the US military), and so we can trim those numbers. let's presume Starfleet has about 1/3 officers, about 250K total staffing, 4 years average as Ensign, and 1/3 of the officers are ensigns... we get 84K officers, 28K ensigns, 7K per year graduating, and a 4 year program at the academy, for a total enrollment of probably 40K cadets. It's a LARGE university, but note that the largest 10 in 2012 run 44-60K students enrolled. Assuming a 1:50 instructor-student ratio, I'd expect about 1000 instructors, and probably 1000 enlisted support staff as well, for a 2000 member staff. Plus probably another 1000-2000 support civilian staff.
Now, presuming technicians normally take a 2 year academy program, and have an average of 5 years service, and are half the service, that could push up Starfleet's academy enrollment quite highly. We hear Chief O'Brien speak of his time at the Academy in several episodes.
Let's assume a 50% Enlisted rate, with an average 60% of them being working technicians, and averaging 5 years until they move on or up, and 50% of officers are ensigns serving for an average of 5 years before promotion or leaving, this means 30% x 2/5 are in technician school; likewise, 25% x 4/5 are in Officer programs, and using a 400K personnel list, that means 80K officer cadets, and 48K are in technician training. 128K would be large for a single facility - about twice what we see for any campus (Miami-Dade University is about 80K on campus, but not all are enrolled; that includes some staff on campus, and a significant fraction of spouses of enrolled students, etc.)
If we instead presume a 1/3 officer, 1/3 of officers are ensign, and 2/3 of enlisted are working technicians, with 6 years for Ensigns and 6 for techs, and 4 years academy for ensigns, 1 for techs, and 400K persons in Starfleet, 44K ensigns, with a 2/3 replacement enrollment rate, about 30K Officer Cadets, and about 44K technician recruits... for about 70K.
One thing we do know - the Academy is a multiple campus system. There is a remote campus at one of the gas giants - not for classwork, but for field activities of various kinds, including the flight range. The Extended Universe includes a Martian campus, and one outside the Sol system, as well.
In the earlier TOS era, we know of 12 large starships. We can presume that there are 2-3 support vessels per large ship... and so probably 50 ships in the fleet, and maybe 3000-4000 members, and so graduating classes of a under 500 per year would be sufficient.