I use pillowcases. I run a line of parachord through the "overlap" area to make a drawstring. works great. I use 6 bags total. one for each leg, one for the loins and backstrap, and one for "everything else" - neck meat, heart, etc.
I imagine that you are correct about game bags one can use for elk, since these may be a lot smaller than moose. The moose quarters are just too big to fit in pillow cases.
Do you debone the elk meat, and butcher it as soon as you get it home, or do you hang it in a dry and cool place for a couple or days before butchering it? In here most people hang the entire quarters (bone and all), and then butcher and freeze the meat. Moose back straps and tenderloins are quite large, but those I immediately cut them to steaks when I get home, and take care of the rest the same day or day after. Aging the meat makes it tender, and that's the preferred method with most people. I just don't have much free time to do that, and hang it maybe a day or two.