If you're lazy, you can buy a K98that has been converted to .308 (Israel made many conversions and they can be had for the $350-$500 range. There are also some .308 conversions of Lee Enfield No. 4s around, these tend to fetch a premium of $800 or so. There are Lee Enfield No. I MK. III's in .308 (Indian) but these are harder to find. P&S Militaria has several Norwegian
K98s that have been converted to 30-06. These rifles aren't cheap either, but they're in great shape.
The SwissK31, from what I hear, shoots very well and is mighty accurate. Brass and dies can be had from TradeEx but it's not cheap. They use .308 bullets so at least that shouldn't be a concern. Would probably make a good deer rifle.
If you're looking for that classic look, the Lee Enfields might do the trick. Brass and ammo is available at pretty much every small shop around the country and .310/.311 bullets are usually available. I prefer the open sights of the No. I MK III to the peep sights of the No. 4 but that's personal preference. They can both be had from $300 to $500.
For Mosin Nagants, brass is harder to find but good quality stuff can be had, at $100 for 100 pieces last I checked, from Hirsch Precision. It's 7.62x53 but it's what you need. Bullets are standard .310/.311 (same size as the .303 Enfields). Accuracy is acceptable, maybe not the greatest but good. The sights on these aren't so great in my opinion, and they may not be up to hunting requirements but I'm sure you'll find many an animal downed by these.
Perhaps the best would be a Swedishmauser, either the short barreled model M38 (light, handy rifle) or long barrel M1896. The 6.5x55 cartridge they shoot is very accurate, recoil is pleasant, and the 140grn bullets have taken plenty of deer. These guns tend to be very accurate and in good condition. Prices tend to be cheaper at $250 to $400. Brass and bullets are usually available, as are dies and loaded ammo. You have to like open sights though.
Enfield Pattern 1914s or Model 1917's use a mauser type action and tend to be very accurate. The P14s are in .303 and the M1917's are in 30-06. They're long and heavy and thus may not make great hunting rifles, but if peep sights are your thing then these would do the trick. Prices vary from $250 to $500 or so, quite a bit of variation in these for some reason.
If you're not adverse to tracking down, or making, brass/bullets for other types of rifles, like the FrenchMas rifles or the old Kropatschek rifles than the world is your oyster, but the above mentioned rifles tend to have easily available ammo and brass.