According to the HITS calculator on the Hornday website, my shooter program, and a 175gr bullet, you should be able to effectively take a medium size game at 700 yards. Hornady defines "Medium size game" as:
A rating of 501 to 900 applies to bullet/cartridge combinations that are applicable for medium-sized game such as deer, antelope, black bear, and caribou, or game weighing 50 to 300 pounds.
So, a .308 will do it, as the HITS factor I came up with is 762 (just about the middle of the road). Yep, it will do it.
The next portion of, "Will my gun caliber do it?" is shot placement. Can you, the shooter, place a first round vitals hit at 600-700 yards? For me, Yes I can. First time, every time, followed with a second round hit every time.
For my gun though, I went 300 winmag as I wanted elk stopping power at 900-1000 yards. Can I do that now? Nope. But I wanted to, so I wanted enough gun ahead of time. Just me though.
On paper, the .308 is inferior to just about every cartridge out there except the 7.62x39 and the 22lr. That said, I have seen first hand, a Model 99 .300 savage dropping a 3x3 buck at 450 yards. I have witnessed a .243 winchester blowing the shot.
Long story short: Yep, the .308 will do the job. Can you do your part and practice to get ethical, high percentage, first round hits in the vitals.