Saiga made some rifles in .35 Remington, and there's a .pdf of a book from D&S from 2000 that describes building one from a flat, opening the bolt face to .460" and sectioning and widening a stock x39 magazine.
The .35 came out in 1906 and is a low-pressure cartridge at 33,500 CUP. It can push a 200 grain, .358" bullet out to around 2000 fps, which gives it quite a bit more punch than the 7.62x39 despite the lower pressure.
A 7.62x39 cartridge is about 2.2" long, the .35 is 2.53", and a .308 is 2.80". You could use a .308 mag with a spacer... or if you handload, Hornady makes some spitzer bullets that might make up most of the difference. The old P-17 Enfield had about the same amount of cartridge slop in the magzine (It was originally designed for the very long .280 Ross), though the idea of cartridges slamming back and forth with recoil makes me wince.
Reamerrentals.com has .35 Remington reamers for $26, with another $5 for the headspace gauge.
Inexpensive true .358" barrels seem to be out of stock everywhere at the moment. Googling around shows some Marlin 336s in .35 Remington were .359 or .360"
Numrich has .357 carbine barrels for $19.95, depending on how you feel about pushing a .358" bullet through a .357" barrel. Izhmash and Tula didn't worry much about that sort of thing...