All we can do is speculate, and infer the size. DSN represents one of the best shows to cauge size from.
From Memory they used elements of 3 fleets to form the task force assigned to retake Starbase Deep Space Nine. And that wasn't complete and it numbered ~650 ships.
So it is reasonable that the number of ships Starfleet were able to call upon was 5000+
Of course that doesn't mean that Starfleet normally operates that number as some have already mentioned many of those ships could have been recommisioned after they where put into mothball storage.
Being kep in reserve should they be needed for an emergency.
Though one thing to bear in mind is any recommissioned ships would still need crew even using cadets and the reserve activation clause it might leave ships with diminished crews.