I have two .308 Savage Model 10 Law Enforcement models and they both shoot under 1/2 at 100 yards with the right loads.
One, a 10 FCP-K, averages 0.489 and the other, a rebarreled old 10 FP, averages 0.426 for all the groups they have ever shot. Some of the loads barely shot under 1 inch but there are lots of loads that broke the 1/2 inch mark. The best 25 loads average 0.314 for one and 0.282 for the other.
My shooting buddy also has a Savage 12 LPV in .308 but it is new and he is just getting used to loading for it. It has a 30 inch stainless steel barrel so it is a bit different than his Savage Model 10 FCP-K whose best 25 loads average 0.325.
So far, his new model 12 averages 0.397 for his 25 best loads but his hasn't yet shot any bullets heavier than 155.5 grains. So far, the loads averages have improved each time he shoots a heavier bullet. I predict it should do great with 168 or 175 grain bullets.
Unless the barrel on your .308 Remington 700 is already shot out I wouldn't look to change the barrel before I tried improving the trigger.
It takes some doing to shoot out a barrel.
If you reload for it, since you can adjust for wear by seating the bullet out further as the throat wears, you can adjust for throat erosion and maintain accuracy. I have heard the 2000 round barrel accuracy limit for .308 from the local LEO Swat team members, but they are required to shoot factory ammo, Federal Premium Gold Match 168 grain SMKs, so they can't seat their bullets out deeper than the 2.800 OAL that the factory ammo comes with.
My first Savage 0.308 10 FP had to get a new barrel after 6300 rounds only because the deep original chamber plus 0.090 throat erosion had me seating bullets out further and further until I couldn't maintain consistent neck tension and the accuracy suffered over the last 300 rounds. Up past 6000 rounds, I didn't see any loss in accuracy but the last loads were seated out at 2.970 OAL. But I kept constant watch on the chamber depth and the throat erosion. I only saw deterioration over the last 200 rounds or so.
The replacement target barrel upgrade that the Savage factory performed actually shoots better than the original barrel ever did and Savage only charged $427 including shipping and returned the rifle in a month completely refurbished.
Try getting that turnaround from a local gunsmith.
From the .22-250 perspective, I have a Remington 700 SPS that I switched to a Timney trigger adjusted to a 2 lb pull and a Bell and Carlson Medalist stock and it now shoots well under 1/2 inch at 100 yards. The factory trigger was gritty and the factory stock flexed too much. It averages 0.474 for all the groups it has ever shot (including with the old trigger and stock and with bullet weights that are not suited to its 1:14 twist) and its top 25 loads average 0.225 with bullets in the 52 - 53 grain range and lighter. All were shot with the upgrades. I shot the Remington 700 today in 5+ mph winds with 53 grain SMK flat base bullets and averaged 0.263 for 17 groups.
If you are shooting your .308 Remington 700 with the factory trigger, you might be surprised how well it would shoot with a replacement trigger. Most drop in.
Depending upon your factory stock, you may or may not get a improvement with a replacement unless the stock does not fit properly or flexes like my "Factory Tupperware" stock did.
My shooting buddy also has a new Savage 12 LPV in .22-250 averages 0.400 for all the groups he has shot to date, mostly with 52 grain match bullets. His best 25 loads average 0.300.