Minecraft wiki entry for Organization, associated text: Build a large room for all your chests. If you are tight on space, you can embed some chests into the floor or make chest "shelves".
Chests cannot open with a block on top (except transparent blocks like torches, glass, slabs, and other chests). You can place upside down stairs or half slabs to change this.
If you want single chests side-by-side without them combining into a large chest, then place them while sneaking to prevent them from joining.‌[Java Edition only]
If you have room, you may want to place signs or item frames next to, above, or on the chests to label them and make categorizing easier. To place a sign or item frame on a chest, place it while sneaking.
If you are tight on space, you can put shulker boxes inside chests for more efficient storage, although looking for one in the End early on in the game is impractical.
Make your storage facilities sturdy and impenetrable. Don't forget to light up the room so mobs can't spawn inside.
Cats love to jump onto chests, which prevents the player from opening it. Make them sit down if you let them into a storage room.
If you have a lot of iron ingots, gold ingots, wheat, lapis lazuli, snowballs, redstone, emeralds, clay, diamonds, bone meal, dried kelp, or coal, craft them into their respective blocks to allow for more compact storage.
Different kinds of logs, wooden planks, saplings, wool, leaves, slabs, stairs, sandstone, mushrooms, tools, food, seeds, music discs, stone bricks, and fungi do not stack. Keep this in mind when organizing.