Yup.
You can order a replacement barrel blank with a .308" bore, and have it externally contoured to match the dimensions of either a 1903 or 1903A3 barrel, then have it chamber reamed for .308 Winchester/7.62mm NATO. If the rifle was originally purchased or papered as a Curio & Relic, it will lose that status once rechambered for a non-issued round.
Cutting back the shank of the original .30-06 1903 barrel and rechambering it for .308 Winchester would make for a shorter barrel once all was said and done, so the front sight would be closer to the handguard/stock and the rear sights, accordingly. It would stick out like a sore thumb to somebody who knew their Springfields.
The .30-06 length box magazine is much longer in the 1903 than it needs to be for the .308 Winchester. So when loading the 1903 up with .308 ammo, it wouldn't hurt to seat the rounds fully against the back of the magazine.
Chamber inserts aren't worth the powder to blow them to hell, BTW.