Is it possible to consistently hit targets at these distances (300-1,000 yards) with a semi-auto .308, given that I would be using match-grade ammunition?
Yes.
What are the key factors that would contribute to becoming skilled in this discipline with a semi-auto rifle? (e.g., barrel quality, trigger control, optics, training, etc.)
While I have a clear bias for one particular manufacturer, there are a few decent large frame gas gun options on the current market (especially if you aren't expecting it to pass full-blown military acquisition test), so unless you completely ignore user recommendations you'll probably be fine. That said, functionality is a critical aspect. If the gun has a stoppage during a course of fire, it's going to be sub-optimal for performance.
While barrel quality is a factor, the performance is what matters. A good gas gun will average between 1 and 1.5 MOA for 10-shot groups. An exceptional gun will be 0.75-1.0 for 10-round group averages. This give very little leeway on small targets that you'll encounter in a PRS-type event, but 800-1,000+ is definitely do-able on a realistic target in real-world conditions.
While it's possible to get good results with a no so great trigger, a decent 2-stage trigger in the 4.5-5.5lb area is helpful. I don't recommend going under 4.5lbs as your probability of having an unintentional bump-fire event rises significantly the lower the trigger break weight is.
I like a flat bottom section of the stock to improve rear-bag use. I don't mind slope in the bottom flat, as i do use it for fast elevation adjustments, but there are very good arguments for having a flat bag rider that is parallel to the bore.
Some projectiles and cases are not designed for semi-auto use. I have seen problems with polymer tips getting damaged, and in some instances break off and shut the gun down. I have seen primers pop out while not at high pressure. I pretty much only shoot factory ammunition, and I have had many that are simply not suitable for gas-gun. I have had acceptable result with Berger, Federal, Winchester, Black Hills, and non-poly tip Hornady.
Recoil management is the name of the game with large-frame gassers, closely followed by trigger control. There is more "lag time" in the primer ignition sequence than with striker-fired bolt guns in general, which can make trigger control a greater impact to group dispersion even if the pull is identical. Working with a pistol, specifically one-handed precision, is a great training tool for trigger control. Triple points if you practice with a long double action trigger. Controlling the recoil and building a durable shooting position will help greatly in maintaining good groups for several reasons, but it also allows you to flex the benefit of the system: fast follow-up shots. Gas gun shooters find bolt guns "easy" and bolt gun-centric shooters find gas guns to be more difficult. Of course there's the whole difference in application and perception which can influence those feelings, but it's definitely easy to transition from gas gun to bolt gun.
Optics, mounts, bipods, ballistic calculators, rear bags, and that sort of equipment are pretty consistent between bolt guns and gas guns, so there's decent info out there on specific items. I do tend toward 1.5" or higher optic mounts for ARs, as lower than that is almost guaranteed to put your head in a bad position and will limit objective bell diameter. I do like christmas-tree reticles for gas guns, but I like to shoot things between 200 and 800 really fast, and that requires holding in a lot of situations. Not really the same thing if you are benchrest-style shooting focused, in which a simple reticle may be preferred.
There aren't a lot of large-frame focused classes out there, but you can find them. Veracity Solutions (Scott Peterson) put one on not too long ago, and I'm pretty sure he said that he was going to do more of them. Good precision rifle instructors should be able to give specific guidance and coaching to a gas-gun shooter, but you might want to check first.
How much of a challenge is it to manage recoil, follow-up shots, and maintain accuracy with a semi-auto compared to bolt-action rifles at long range?
Technically, the longer the range the less good at recoil management you have to be, as the time of flight gives you time to get back on target. The real challenge is at 100-200 where you will need to fight to control recoil to stay on target and "check your work through your sights". I use a pretty standard bolt-gun shooting position: very square behind the gun, stock as close to my centerline as possible. Learn to shoot the gun with a neutral-load or touch-load at most. I do use pretty firm rearward pressure with my firing hand, but that's something that you will need to work out on your own. Body position and natural point of aim are big contributors to good recoil management.
