For best accuracy in that price range get a Savage or a Remington 700 SPS. If you get a SPS make sure it is the newer version that is pillar bedded.
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FNAR
FN quality, semi auto, and 20 round mags.
If I were in the market I would look at the Ruger American 100% American made, can't go wrong. Can be found a walmart of all places.
I wish i could find one in my price range.
Budget considered the Savage series makes a hell of a rifle that will typically keep up with guns costing twice. Get yourself some good glass (around $200) and strong base/ring set. That and some ammo will blow your budget.
No super popular but HOWA makes one hell of a rifle for the price. Made in Japan of all places. I have a 20 in bull barrel Howa that can rival a Rem 700.
Which savage series? There are so many that I don't wanna buy the bottom line one. I would rather spend the extra money and get a decent one.
This Savage can be found right at your price minus optics http://www.savagearms.com/firearms/models/
It might be a little long & heavy for carrying around during deer season but would definitely work for varmints and bench shooting.