Best sniper rifle 308

An actual purpose-built sniper rifle would be chambered for the 7.62x51 NATO cartridge, rather than the civilian .308 Winchester. That aside, "best" is subjective, as is "civilian"... the Remington 700 is a "civilian" weapon, but the M24 and M40 sniper rifles are variations of it. The Winchester Model 70 is another civilian rifle which has seen adoption with some military forces. So, laws on what's legal in your jurisdiction will play a role here, as well as what type of environment. It's perfectly legal to own something like a Stoner SR-25 in most of the US... whether it's the best or not depends on the environment (it would be more beneficial in a built up area rather than an open area where targets could be expected to be out at further ranges), and also the level of training the operator has... while the SR-25 is a very accurate weapon, it's designed for a highly trained professional force, whereas designs such as the Galatz, SVD and variants (to include the Chinese NMD-86 variant, which was also available in 7.62x51 NATO), PSL, M76 (also available in a 7.62x51 export variant) were designed for conscript forces with a lower level of training, and thus, suits those forces much better.

Now, if you're looking at an open operational area with longer ranges, you'd want to go with a bolt action. The aforementioned Remington 700 and Winchester Model 70 are both Excellent rifles, which are common and have a high availability of parts and aftermarket accessories for them. The Ruger Mk. I is also an Excellent rifle. In any instance, a free float bull barrel and chassis stock which keeps all external pressure off the barrel would be desirable.