There is a lot of mention of "quarters" here. Almost none of them are actually referring to what a butcher calls quarters though. If you want to "quarter" a moose, you need a good butchers splitting saw ( ideal ) or the chainsaw mentioned previously. Trying to split the spine lengthwise accurately is no easy task. It is much easier to be precise if the moose is hanging up. But hauling a whole moose out of the bush to hang it up is just silly IMHO.
What we do is gut the moose, lay it on its side, and skin out the top side. Keep this skin clean as you work. Using a knife, cut off the shoulder and front leg. It is not attached with anything a knife can't handle. Bag the front leg in a cotton bag. Extra large pillowcases are almost ideal. Cheese cloth is way too porous and lets in too much dirt. Set aside on some clean grass or leaves, or better yet hang it up to cool. Separate hind leg from pelvis and work up and around, close to the hip bone and spine to keep as much of the meat (round steaks) as possible with the hind leg. Feel for the ball joint by moving the leg and cut the small tendon that holds the ball to the pelvis with the tip of your knife. Bag that hind leg as you did the front. You now have a legless carcass on the top side. Cut the ribs off on that side, using a saw, or hatchet. Saw is better, makes the ribs less dangerous since there won't be as many sharp shards to handle. Cut them so that you don't damage the rib steaks, perhaps 6" from the centre of the spine. Roll carcass over on the clean hide to the other side, skin that side, and repeat removal of front leg, hind leg, and ribs. Both rib slabs will fit in one bag. Split backbone horizontally between second and third ribs. Bag the loins and pelvis. Remove head if you haven't already. Bag the front backbone/rib steak and neck piece. that't the biggest , heaviest piece by far. You will have 7 or 8 bags of meat, bone - in, in nice manageable pieces just a good size to put on a packframe. Four legs, ( Not "quarters"!) the two spine pieces, one bag of ribs, and the head if you keep it. The advantage of cutting the moose up with bone attached is cleaner meat that can be hung to age without shortening. Backstraps removed by the "gutless" method will shorten and become tough, and no amount of hanging to age will yield as nice and tender steak as you will get by hanging "on the bone". No quad or winch or power equipment needed, just this: a good packframe, some cotton bags, a skinning and perhaps a slimmer deboning knife, a folding saw or hatchet, some cord to hang the meat to cool, good boots and some muscle. Have done it many many times. Note this method will not be OK in BC, where proof of ### needs to stay attached to the carcass.