Short answer:
No, don't shoot moose at 600-700m with a .308. Ballistics data for huge rate of bullet drop greater than back to brisket distance within 600-700m, and realities of wind swirls and gusts at that distance, and shooter errors in reading wind and distance and mirage, and trigger pull, all stack up to high probability of wounding the animal and not retrieving it.
Long Answer:
On Hornady's website I chose their .308 178gr ELD-X Precision Hunter ammo, (what looks like their best long range hunting round in .308 for big cervids), and clicked on their ballistics calculator to see what is happening at 600-700 yards (yards not meters). For settings I left the atmospherics as-is, except I added a 10mph wind at 90 degrees.
Below are some summarized results from Hornady's specific test computer program, calibrated however they calibrated it. They used a muzzle velocity of 2600 fps - guaranteed your rifle will perform differently. I do alot of reloading and using chronographs and a Kestrel, and shoot steel at mid-range distances, and its never the same as a published data table or ballistics calculator output - you have to derive your rifle and ammo's performance data. And that wind, oh the wind! It swirls and switches continuously.
From Hornady Ballistics for the .308 178gr ELD-X, standard input data except for 10mph wind at 90 deg:
At 600 yards:
Velocity: 1737
Energy 1193
Bullet drop: 98.4 inches
Wind drift: 27.4 inches
At 700 yards:
Velocity: 1615
Energy: 1031
Bullet drop: 147.2 inches
Wind drift: 38.7 inches.
The terminal velocity in Hornady's results is possibly adequate for bullet expansion, but some will argue it may need to be higher?
You will need to confirm your average muzzle velocity, in order to plug in your data into a calculator. Keep in mind that as your ES gets larger, the "average" becomes more and more meaningless for a clean hit. Unless you have a small ES, you will not have a precise idea where that cold bore shot is going to hit.
And you will need temperature and barometric pressure input data with a Kestrel or other device, and test shoot out to the same distances, while estimating average dominant wind forces along the flight path. The Kestrel's wind reading where you are standing will not necessarily be the correct windage input - you need to read the trees, grass, water, mirage, etc, and look for gusts and switches. F-class shooters hit or miss by watching the wind flags and mirage closely on the range - do you have these wind reading skills?
I find ballistic calculators, before tuning, are not accurate out to distances like 600-700 yards for my rifle and handloads, despite my control of input variables such as tested chrono speeds with my handloads, and using a Kestrel for the day's atmospheric inputs.
With your distance ranging skills and equipment, do you know how to dial or hold over between 600 yards and 700 yards with a bullet drop difference of 98 to 147 inches? Every few yards of distance estimate error makes a big difference in point of impact. That is a 49 inch difference in drop. For context, moose back to brisket measure is roughly 30 to 36 inches.
With your wind reading skills can you really estimate windage dialing or holding between 27 and 38 inches in that 600-700 yard interval for the perfect 10mph 90 degree wind? The wind likely will not be that exact, so how will you correct for wind drift to avoid wounding the moose?
I am not doubting your skills and equipment. What I am saying is that based on the enormous rate of bullet drop and effects of wind on the bullet at those ranges, even the best shooters can easily miss the heart-lung vitals zone, leading to a non-fatally wounded animal that is going to escape and suffer a long painful death later, and the meat will be wasted.
Technically do-able with a .308? - yes. Should you? I say no, not safe enough to take that shot. Move much closer (well within 400m) for a higher percentage clean kill. I suggest test shooting on paper to determine your point blank range for +/- 3 inches, and then calculate your risk for accurate dialing or holdover or windage beyond that distance, and keep it small. If you think you need to dial scope windage to shoot a moose, I suggest that is already too far. If you are not comfortable with your reticle to hold a wee bit for wind with confidence (based on lots of practice), then the scope dial is not going to help with accuracy.
Regarding holding for windage: Target shooters will be very familiar with holding for wind and having the round hit exactly where you pointed, missing the shot and scoring low in the outer rings, meaning your wind call was totally wrong at the moment you pulled the trigger, despite it being probably good a second or two before you pulled the trigger. Wind swirls, gusts and switches alot!