[JJ_BPK]

As you didn't say what your shooting at 600-800 yds,,

Let's assume paper,, because the .308 (7.62x51) starts to lose a lot of punch past 600 yds.

It is and will continue to be used as a sniper weapons round with good success, but as a hunting round, it becomes marginal at that distance.

So, for Paper,, 600-800 yds,, Find a weapons builder and go for something in a Remington 700 action, 28" bull barrel, match action, bedded stock.

In fact, if your going to spend big bucks,, and for best ammo consistency,, I'd tell you to look at something that carries more powder & punch.

Some suggestions:
300 Win Mag
6.5x55 Swed
.338 Lapua

I'll WAG this,, but 5-6k USD should get you started, providing you already have reloading gear??

One big factor in match shooting over longer courses is the ability of your scope to have enough "clicks" to adjust from say 500 to 800 yds.

If I remember,, The .308 sighted in at 200 yds has about 50 inches of drop at 500 yds.. That's a lot of adjusting to move the POI out another 500 yds.. A lot of scopes just can't do it,, so figure another 1.5K USD for a top of the line scope.

OR

Try your FAL.. minute of man at 600 yds,, all day long,, if you do your part..

OR

Get into and graduate from sniper school,, Uncle will provide all the tools you need..

Good Luck,,