I am sure there are several weapons out there that will fit the bill but I am sure that the one that will be mentioned most often is the Remington 700. It is probably the most commonly used rifle for snipers by the US Military as well as police departments. The most common cartridge would be the .308, but is also very capable of accuracy at long distances in 7MM Mag & .300 Ultra Mag.

Another rifle you may want to look at though is one of Savage Arms many variants in the same calibers. I can't say enough about Savage's Accu-Trigger and it's ability to help tame trigger flinch flyers.

To Balt Tigger's excellent answer, I wish to add that the Savage has an easily replaceable barrel, which contains the cartridge chamber. Assuming you trade regular non-magnum standard cartridge head dimensions one for another, or the same for trading one standard magnum cartridge head dimensions for another magnum calibre, one can easily switch to a different cartridge as one might choose after changing one's mind about recoil levels, for example, or to someday replace a worn out barrel! To that end, one may purchase a worn out Savege cheaply, and make it into whatever you want (with the same cartridge head dimensions), even if it would have been a difficult-to-find used gun, but easy to order a replacement barrel for, from Savage. Regards, Larry.

Does the $700 include your scope budget, or is that seperate?

I'll give a specific recommendation, and I agree that .308 and .30-06 are the calibers you want. Redneck?...well, maybe, but if you can't good advice about a hunting rifle from a redneck, who can you get it from? LOL

Remington 700. Best of breed, best in class....you just cannot go wrong with a Rem 700. There are other good rifles out there, plenty as a matter of fact including the Savage, but Remington leads the pack with the 700 and priced right.

Nikon makes the best affordable riflescopes. If you have more, move up to Leupold, but for two bills, you cannot beat the Nikon.

Here's my suggestion.
Remington 700 SPS .308 $496.99
Nikon Buckmaster 3-9x40 BDC Reticle Matte $209.95

Puts you just over $700, prices are from Impact Guns and this is a hunting and beginner target shooting package that you cannot go wrong with. It will provide years of service and if you decide to move on to something else, you can get top dollar for a package like this when you sell or trade.

If a good friend asked me to help him spend his $700, this is what we'd get.