Huge?
In 1945 the United States Navy had some 833 surface warships (out of 6,768 total active ships), whilst the circumstances are somewhat different (we depend on dilithium, we represent 20 unified planetary governments/economies and countless colonies) if we kept things 'proportional' (each member being a United States) Starfleet could easily support 16,660 ships.
Going for the more age of sail/exploration feel of TOS, we can look at the Napoleonic Royal Navy (Napoleonic Britain being our closest comparison as far as industrial, economic and bureaucratic development goes), in 1805 it possessed 103 ships of the line, 120 frigates and 420 sloops-of-war/other minor warships (643 ships total), with the same assumptions as above you'd get a rather comfortable 12,860 - or 4,460 ships if you discounted the sloops.
If you want huge senseless numbers in Star Trek the only real example I can think of is the Mars orbital ring from Picard S1, before the destruction of the Martian orbit/surface, which comes out to about 63 million starships. ~7,500 starships is perfectly reasonable assuming a late TNG era fleet size of 20,000-40,000 (which is, iirc, broadly accepted)*.
*Keeping in mind the Federation is a polity with far in excess of 900 billion citizens at that point.