Maybe want to confirm the terms that you used?? The "bore" is the hole drilled down the barrel. You say it measures .308". The bullets need to seal tight to the bottom of the grooves - in a No. 4 the grooves are typically .004" deep - so grooves on each side, that ".308" bore" barrel is asking for .308" plus .004" plus .004" equals .316" bullets. So the barrel sounds like a .308" bore and .316" groove barrel. That would be well worn, about .005" over original spec, but not unusual to find, at all! Unless you have pulled the Winchester factory bullets out of the cartridge case, you are not likely measuring at the bullet's widest diameter.
You can use a pure or soft lead "slug" and slug your barrel - take a ball of soft lead about .325" or so and gently tap into the muzzle, and perhaps a few inches into the barrel. Then use a cleaning rod to tap it back out. Your "slug" will now have reverse impressions of the grooves. You can measure them with a calliper - unless you have a five groove barrel!!! Accurately measuring a diameter from 5 grooves takes some fancy math and some special tooling that few of us have.