My main question would be...a full-size pistol for what?
Self-defense, hunting, target shooting, etc?
Good info to know.
Also, I assume that money must be very, very tight if $300 is stretching it a bit. If that is the case, I'd also take into account ammo cost. Some of the aforementioned .22s might be viable options, since you can get 500 rounds for under $20. $20 won't even buy you 100 rounds of 9mm, the cheapest center fire round in most locations.
If not a .22, and if the gun will be used for self defense (again, on a very tight budget) then I'd also suggest a used .38 revolver. S&W Model 10s can still be found for under $300 if you look. Revolvers from Taurus, Ruger and Charter Arms (some used, some new) would also fall within your budget. Yes, .38 special is more expensive than 9mm. But if you will be using this gun for protection, I assume you'll be stoking it with good quality hollowpoint ammo. And if you are going to rely on it for protection, with an auto I wouldn't trust it until I had run
at least 3 full mags through each mag you would load with hollowpoints, without a glitch. Preferrably twice that many. And that step comes
after running many more rounds of cheap range fodder through those mags without incident. So you are talking several hundred dollars in ammo just to verify reliability.
With a revolver, a few cylinder fulls of your chosen defense ammo to verify POA vs POI, case extraction, ignition reliability, etc are all that is absolutely necessary (after a hundred rounds or so of cheap ammo to test function). So initially, the ammo cost outlay to verify reliability with a .38 revolver will be lower. Over time, a 9mm auto would be cheaper to feed.
One other off the wall option. If this weapon will be strictly for home defense, you might consider a pump 12 or 20 ga shotgun. New Maverick 88s can be had for $225, last time I checked. And there are plenty of used, sub 20" equipped pumps floating around for under $300.