A.,
Just curious, when you say "need" is there a specific reason other than sentimental reasons that you need to keep them? The whole sentimental attachment can be very powerful and sometimes lead us down a very cluttered path. Your kids will always "get" more things/animals, and sometimes we do need to be judicious about what we keep.
The easiest thing would be to store them in a box, but then you can't see them. You could get a curio type/hutch that has glass doors. Like some people would use for china or good dishes. but those can be expensive and take up room.
I did get a shelf from Michaels that was about 5' long and I stained/laquered it, then attached hooks to it. I hung it high enough it would be over a bed/dresser. As I have a Hard Rock Cafe Bear collection. The bears all sit on the shelf and my kids can use the hooks for coats or purses/bags/backpacks.
My shelf is now full (no more bears???)....and it's not extremely wide so a large animal would not fit.
If it's for their room, you could attach a netting type fabric across the ceiling. Like when you suspend balloons. It's bigger than the little corner nets...that don't work. You can double layer it for strength and be sure to attach it in with something sturdy through several folds. you still won't be able to put a bowling ball in it or a hundred animals but it would get them off the floor and something for the kids to look at night. you could even put a decorative take periodically thru the middle areas so you had little "scoops".
Then when your boys get older, if they want to "lighten the load" and don't know how to choose. They can make a list of for each animal of "Why they love this one" the one's with the fewest, least compelling, win a trip to a garage sale or donation center.