Yes or No

Yes. The Army Remington sniper rifle was built on a long action. It was not because of concerns of OAL but so they could swap barrels and bolts and have a .300 Win Mag rifle.
 
A 30-06 is a long action, and will have the same bolt face as the .308, so to make that work you would have to re-barrel, and that would be it. There are places out there that make longer magazines for short actions, Wyatt, Seekins, Badger, etc. May be a cheaper option than a re-barrel.

R.
 
Yes. My first .308 target rifle was built using a .30-06 M70.

Since you want to feed from the magazine, make sure that it feeds well. In most staggered box magazine designs (such as the Winchester box magazine on my rifle), the angles of the feed rails need to be closely matched to the case body taper. A single cartridge feeds quite reliably out of my magazine, but when you put in several .308s, the small difference in case body taper between them and the .30-06 results in them not sitting straight. It ought to be straightforward to fix this, it's just something that hasn't mattered to me.
 
You can also buy a .308 follower spring and put a spacer to achieve what you want with the longer action. 30-06, 270, 280, 338-06 - basically anything with the 30-06 family will allow you to build a .308 Win. Just out of curiosity what length OAL are you thinking of loading. I have never had an issue with bdl type rifles for OAL in the 700 family. I only had OAL issues with the detachable magazines. Unless you have some really long bullets you could just build a normal small action and single feed the ones that are too long. I was always able to get good accuracy with the 155/168/175 gr bullets and have them feed from the small action mag.
 
Remington factory .308 Win chambers tend to be quite long-throated. If you want to shoot match bullets (which have much longer noses than hunting bullets), it can be impossible to get anywhere near the lands while still being able to feed from the magazine. I helped a friend with a 700VS develop some ammo, he was seating his 190 Sierras to 3.0xx" or 3.1xx". They they sure shot well, though they were quite different looking from a mag-length .308! They wouldn't fit in a magazine, nor would they fit in .308 ammo boxes (had to use .30-06 ammo boxes).
 
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