Calling the rifle a sniper gun has already been covered so I will leave it alone.
300 yards is not long distance any longer, it's not even medium distance. Most hunters even with minimal abilities can successfully hunt out to 300 yards. Years ago 1000 yards was the pinnacle of long distance shooting, these days 1 mile is the new 1000 yards. With the advancements in rifle production and barrel fabrication over the past 20 years 1 MOA rifles are the normal even from entry level bolt rifles. Actually, if an entry level hunting rifle can't shoot 1 MOA these days the rifle usually fails in the market.
All that said, the above suggestions for buying a Savage rifle are good suggestions. Add to that their excellent AccuTrigger and you have an inexpensive solution that will surely do 3" @300 yards if you do your part.
Suggestions under $700 (without glass) and I'm a 30-06 fan
Savage Model 14/114 or Model 16/116 (maybe the Model 111 if you can find a sale)
The Howa 1500, my 2 main bolt action hunting rifles are Howa 1500's in 30-06 and .223.
The Ruger American is getting a lot of good press, it's worth a look.
The Thompson Center Venture is a very accurate rifle and for some reason overlooked.
The Mossberg ATR is a fairly new bolt action rifle that is getting very good reviews.
Of course the Remington 700 is a proven platform and can be found under $700 if you look around.
The Winchester Model 70 is also an outstanding choice but these days it's out of your budget range as are Tikka, Sako and most Vanguard rifles.
I guess what I'm getting at, like I said above, there are many good low price rifles out there today that will do everything you want to do and maybe even more...