We know canonically from WHMHGB (TOS) that ships can and have traveled all the way out to the galactic rim, and have explored all the way inward to the Great Barrier around the galactic core. Even if we take Picard's "8,000 light year" figure as a solid sphere of 8,000 ly in diameter, the Federation "sphere of influence" would be 268,082,573,106 cubic ly (rounded to the nearest ly). Using the commonly accepted figure of a 20 ly "sector", that would be 33510 sectors (again rounded to the nearest sector). For Starfleet to maintain any sort of presence in just 1/3 of them would require a fleet of 11,170 ships at 1 ship/sector. That assumes that all ships are always out in the field all the time, which does not allow for ships to be drydocked for repair/refit, crew rotation, or any other function. And again, that assumes the Federation is a single, contiguous spherical area. Spreading the Federation out over a larger, non-contiguous area would increase the amount of ships needed to maintain such a presence, as some ships would be always in transit, and thus "out of position". Actually, it's the other way around. Warships tend to have bigger crews than civilian vessels because of the need for crew redundancy in case of casualties during combat, plus the number of highly specialized functions that only occur on combat vessels.