There is always the "Quasi-Ar" design of the "Bad News Semi Auto" which comes in .300 Win Mag, or in .338 Lapua it is available
here
Although for a precision rifle, I had much rather have a magazine fed or even single-shot bolt-gun.
If you are wanting a .338 Win Mag, the only option I know of is to get a Browning BAR, and accessorize it however you desire. Most accessories for it would have to be custom one-offs though and the price would be preventative. I have heard rumors of them being converted into .416 Taylor, which despite commercial ammunition not being available, it would make for a good game rifle.
McCann Industries converted M1 Garand rifles into .458 Win Mag, but I think they no longer perform the conversions.
If size is not a problem then the 17 lb Vigilance Rifles offers a semi-automatic 5-shot rifle in .338 Lapua, .375 Cheytac, .408 Cheytac, or .505 Gibbs. Not by any means handy, but a SA none-the-less. They are available
here
The Bad News by "Only Long Range . com" seem to be the closest rendition of an AR, when it comes to big-bore semi-auto rifles. If I need a long range, I will stick to my Barrett. If I need big-bore I will stick to a double-rifle. But for a true hunting AR-15 or AR-10 design, there is a number of good choices for a hunting rifle. For the short action (AR-15) I prefer the .50 Beowulf as a hunting companion. I know it will take anything I ask of it, especially when loaded with my 350 grain Solid Brass Spitzer rounds marketed as a "Dangerous Game Load". They certainly pack a wallop on most North American game, and provide for quick follow-up shots. Likewise, similar concepts including .458 SOCOM, and .450 Bushmaster provide similar muscle to bring game down inside of 150 yards.
Perhaps the best (In My Opinion) hunting style AR-10 chambering is the .338 Federal. I have used this little cartridge pretty extensively and have nothing but good results from it. It recoils only slightly more than the .308 but packs a larger diameter and heavier bullet. A 210 grain bullet at 2630 feet-per-second with a .338" diameter will ruin just about any animals day, provided you do your part. In a pinch brass can be made from .308 brass, and projectiles come in many different flavors and toppings. Loaded in the AR-10 , I would feel confident to hunt any game any in North America, including having protection against the big bears. But this is just my opinion and yours may vary.