It probably could but what's the point? The short ackley is shorter than the 30-338 and it headspaces on the belt so you would probably get away with it but it certainly would not be a practice I would recommend.
I have, on a couple of occasions, fired 300 Winchester mag ammo through a 300 Weatherby rifle. The result was a piece 300 Weatherby brass with practically no neck. Accuracy was good enough to take big game animal at anything under 100 yards so it could work in an emergency situation. Still, it can't be considered to be a good idea. GD
If the ackley is a short mag like the WSM then NO.
If you are thinking of fire forming. Then it has to be the same length cartridge. I have a rifle in 6.5x57mm I can not buy factory loaded ammo for it, so i buy 7x57mm brass neck it down in the 6.5x57mm sizing die then shoot it. which fire forms the brass to my chamber. Some people will do this then only neck size the case (Sizing only the neck of the case and not a full length resizing). This works the brass a lot less which will give you slightly longer brass life.
There are no big time headspacing issues. Both cartridges are based on belted magnum brass which will headspace sufficiently well on the belt to work OK in that regard. In fact, it is intended to headspace on the belt though headspacing on the shoulder is preferable when reloading belted cases.
It is worth mentioning here that there were two Ackley short 30s. The No1 was the shorter cartridge which was designed to have a case capacity of approximately 70 grains. Ackley considered this to be the optimum capacity for a 30 caliber and, at the time it was developed, this was likely not far off since 4350 was about the slowest powder commonly available. The No.2 Ackley was a slightly longer case and was, in fact, very similar to the 30-338 or the 308 Norma (closer to the Norma I think). The purpose of the No2 was to allow 30/06 rifles to be rechambered without the necessity of setting the barrel back to clean up the chamber.
The upshot of the whole discussion is this:
The 300 Ackley Short mag (No. 1) could probably be fired in a 30-338 chamber with no ill effects. The No.2 ackley short mag could not since the case is slightly too long for the 30/338 chamber. The No.1 short mag brass would end up a bit too short and I truly don't see that point. Brass isn't that expensive. Just get the right stuff! GD