There used to be available (and may still be) lots of Swedish Mausers in 6.5 X 55. You still see lots of deals on the ammo, and that cartridge shoots very flat with a pretty tame recoil. The 96 model was extremely accurate and well made, as was the shorter carbine (I think it was a model 38). You can buy sporter stocks and dress it up if you want and it won't cost the farm. You MAY be able to get a .308 barrel for it (and you definitely can get 7mm Mauser), but you might want to check with someone like Gunparts Corp first to confirm the barrels that are available. (It's a small-ring Mauser.)
There are Enfields floating around (.303 British) that are still cheap and they have a good reputation for accuracy and dependability. Again, sporter stocks are available and ammo is cheap. I've seen them for $100 and up here in the north. (I owned 2 ... bought the first one for $25 w/case and 20 rds., the second I paid $50 for. Them days are gone.)
If you are from Canada, I'd suggest either Milarm in Edmonton, SIR in Winnipeg or Marstar near Ottawa. They sell all these old bolt actions, and Marstar sells actions and barrels if you just want to build a sporter.
I don't know the US sources very well, but Inter Ordnance in NC is advertising K98 Mausers in 8mm for $100 and up. These can be easily converted to lots of calibers (including .308) with a sporter stock very reasonably. If $100 buys good metal, don't worry about the stock ... sporterize it.
SOG is advertising Turkish Mausers in 8mm for $55, but be warned, re-barreling it may not be easy or even possible ... they apparently used 3 different thread/shoulder combinations (at least). Sporterizing it is easy however ... it's almost the same profile as the large-ring.
If Russian calibres don't upset you, there seem to be Moisin-Nagants from Romania floating around for the cost of a good lunch. Somewhere the other day I saw an add for $40 for one of these but I don't remember where. Check the Shotgun News.
I agree with the Garand suggestions from others in this thread ... if you can find yourself a deal somewhere. That's a fine weapon. It's .30-06, a little heavier than the .308 you were looking for. I'm from Canada, so I have seen deals up here on SA's, Winchesters (both returnees from Denmark) as well as Breda's and Berretta's (surplussed from Italy). I just got my VG - SA, Feb 1943; and my mid-50's Beretta in VG to Excellent arrives next week! (So I guess you could say Santa has been good.)
Hope this helps.