Any bolt action and probably most other 308 rifle, lever, auto single shot, or pump could be rebarreled to 338 Fed.
If I were to make one I'd probably start with a Mauser, like a Yugo 98 or other KAR M98 and just build it from the action up. Making the 308 feed from the KAR 8mm Magazine is not too hard to do in most cases.
You do the barrel work and then do the feed lips. Once you get it to feed smoothly you weld on a new bolt handle, re-work or replace the trigger, and install a new safety (if you are going to mount a scope in the regular position.) Then install the sights and/or scope base.
Lastly you stock it with any stock you like, be it wood, laminate or plastic.
When you are done you can have the rifle you want, instead of one that's "pretty close"