I know a few guys who hunt LR for elk, popular cartridges are .300RUM, .338 Lapua, and .408 CheyTac.
A .308 Win (IMHO), isn't near enough for elk at 700 yards. I have killed a fair number of elk and are a strong animal. My minimum for a 700 yard would probably be a .300WSM, but I would probably prefer something a bit bigger. I like heavier bullets and more velocity for LR Hunting. One of the biggest problems I have seen is that Mr. Murphy is there and likes to work his magic. A bigger bullet with more velocity even on a poorer shot is going to give you enough time for a follow up, as it will do a greater amount of damage.