The .308 is also called the NATO 7.62x51mm. You can definitely get your hands on some surplus, or cheap target ammo without breaking the bank. The rifle is another story.
DPMS and Armalite both make .308 AR style platforms. Give Remington's website a once over, they have a line of AR style varmint rifles now, might be able to snag a .308 there.
However, you said your looking for a cheap hunting/target gun. If you want cheap and .308, try looking for a lightly used gun.
Good luck with your search, and make sure you do your homework. As the guys on here said, theres an AR for literally everything, from .410 shotguns to .50 Beowulf. I just haven't found an elephant gun yet, but I'm sure theres one lurking out there somewhere
"I only regret I have but one life to lose for my country" Nathan Hale
Amendment 2 - Right to Bear Arms. Ratified 12/15/1791. Note
A well regulated Militia, being necessary to the security of a free State, the right of the people to keep and bear Arms, shall not be infringed.