If you have any control over the locks themselves, get them keyed up on the same keys.

Otherwise, sort them by as many use categories as you like (every day, every week, once a month, etc.) and get rid of anything you do not NEED. I cut my keys down to my carkey and house key which is now keyed to open all doors including my french doors.

Reducing the number and organizing them in groups allows you to make a better choice up front. Put the uncommonly used ones away somewhere secure where you can get to them or packaged a wallet or metal case if they still need to be on your person.

But your question is about storage. I've found some of the hard cases for camera memory work nicely with some foam for soundproofing. They also make keysafes which allow you to carry one key and store the needed key near the site, though this is obviously less secure than just keeping the key with you.

I suggest actually visiting well stocked locksmith... they usually have higher quality items for folks who have to deal with a lot of keys.