Boxes have worked fine for me. Cardboard variety.
I fit 4 pretty big boxes in a 2x2 situation. 1 for pork, 1 for venison, 2 for beef. One is ground beef, one is steaks/roasts. Then we have the shelf with some chicken, vegetables, and other random stuff.
The boxes fit perfectly in my freezer so almost no wasted space.
The key for us has been having a box numbering system top left to right is 1 and 2, bottom is 3 and 4. Then, on top of the freezer I have two dry erase boards attached with two sided tape. We take inventory before the boxes go into the freezer and amounts/types of meat gets put on the dry erase boards. This way we know before even opening the freezer what box we're after or if we even have any more of that type. Pull the meat out, update the count on the dry erase board, and you're done.
For venison, we only freeze the backstraps and a couple roasts from each deer. So there's several deer in that box at any given time. I bag and label each deer separately so we eat them in the same order I shoot them. I just use plastic grocery bags for this. Tie the handles and staple a paper tag to it.