If $300 is the top dollar you can spend I'd say the Mosin Nagant is going to be one to look at. Any rifle that is good enough to hunt men with is good enough to hunt deer with.
Yard sales and pawn shops may have any number of more modern rifles now and then for sale too, but you have to look around, and a lot depends on where you are located.
At $400 things open up a LOT more, but $300 is going to be a challenge.
If you find a more modern rifles for $300 in many cases you will also need to buy a scope and mounts. So a $300 rifles probably really is a $400-$450 rifle or more anyway before you are ready to fill the freezer with venison.
If you use a Mosin Nagant you are ready to hunt Bambi as soon as you zero it in with soft point ammo.