Do you mean benchrest shooting as the type of competition or just shooting from a bench?
If you mean 100/200/300yd shooting with 5 to 10 shot groups, forget about the 308. WAY too much recoil for this type of fun.
I would strongly recommend a smaller chambering with the 6PPC the defacto choice (6BR is another gem as is a 222 or 223).
I just put a Shilen Select match prefit on my Savage test rifle in 6PPC and 100yds groups are hovering in the 1's - I even got a few 3rds test groups in the 0's. I have a bit of tuning to do and looking forward to days with less wind but it will shoot if I can drive it.
If you are putting together a rifle with no worry about a set of rules, then I can lend you some info on my rig. There are also some customer rigs on my website that can lead the way.
If you like a 30cal, the 30BR is also superbly accurate and a top choice in SCORE shooting. Barrels seem to last almost forever too.
A savage can be made to be very accurate for a factory based action and I feel the best bang per dollar spent. BUT it may not be fully competitive in SR BR depending on the class you want to enter and the types of rigs you will bump up against.
if you want to have an inexpensive rig that can put them into one hole at 100yds, a Quality barrel on a savage in an appropriate stock and rests WILL do it.
How small that hole is, is the question...
Let me know if I can help.
Jerry