TARGET?
You didn't really define 'TARGET' in your post.
I have a 10-year old Rem 700 heavy barrel in .308 Win. I would define it as a 'target' rifle when competing against friends at the range. But 1/4" groups won't win many competitions.
About the cheapest 'Target' rfle chambered in your caliber is the Savage Long range TARGET rifle. It lists for $999 and is GUARENTEED TO SHOOT sub-1/2" GROUPS with MATCH AMMO, by the factory. Available in .223, .308, .300 Win Mag. This is probably the best buy for the money. And it's GUARENTEED!
Any rifle could be considered a 'Target' rifle, the level of competition dictates the performance required. If you just want to shoot small groups for your own enjoyment, any heavy barreled rifle would probably work. If you're talking Benchrest competition, you're asking for hole-to-hole accuracy. This would usually require a different platfom to build on. In my area, the Remington Custom Shop 40Xs make up over half the entries. Even these require additional work to get into the .1" groups.
Optics (Remember, you asked for opinions).
The old saying for tactical-syle counter-sniper rifles is to spend as much on the optics as you did on the rifle. They are usually machined form a solid piece of metal into a one-piece tube design. This allows for a more rugged housng, which is less prone to damage. They also employ much better optics to ensure greater clearity and light-gathering abilities. Because of the engineering ivolved, the price tags are usually in the $1,000+ range, usually costing MUCH more.
A benchrest rifle sees a lot less abuse. It is usually only carried from your vehicle to the house/benchrest. It can get by with traditional multi-piece tube design, and because you will usually be shooting in daylight hours, it won't require the expensive multi-coated lenses found on a tactical scope.
An inexpensive Leupold can easily handle the chores of paper targets, while usually costing less than $500. There are also several other brands that can be had for even less. For a 'dedicated' paper target scope, I would suggest a fixed 24X or 36X, but this limits you to targets only. A variable power or less powerfull fixed scope will make for a better target/varmint rig.
Good Luck