http://www.minecraftwiki.net/wiki/Chest#Bugs See the list of items here where you can still open the chest.
Liquids: Water and Lava.
Plants: Leaves, Cactus
(With a water source block on top), and Farmland.
Translucent Blocks: Glass, Slabs, Stairs, Monster Spawner, Snow, Ice, Fence, Cake, Beds, Glowstone, and Chests.
Non-Blocks: Torch, Redstone Torch, Redstone Wire, Redstone Repeater, Pressure Plates, Lever, Ladders, Rails, and Sign.
I personally like Fences because you can make a nice checkerboard pattern of chests and fences
The other solution is to place your chests in the ceiling with another block empty above them; nothing can get on top to sit on them.
gaming.stackexchange.com/users/9091/Andrew Wonnacott
You can place transparent blocks such as slabs, steps, or glass one block above the chest.
In fact, even chests stacked on chests will work.
These will allow you to use your chests still, but cats will not be able to sit on top of them.
gaming.stackexchange.com/users/11967/Alexis King
Stack your chests on top of your chests.
This will prevent anything from climbing on top of them, and you can still open all of them.
Either that, or don't let your cats in your stash room :)
gaming.stackexchange.com/users/18337/Ryan S.
Well, you can place a transparent block such as glass or glowstone above your chests or you can use a slab.
If that doesn't look good, just place another chest on top of the first chest, that way, you have extra storage and, your cat can't get on top of either one, unless, there is a block next to the chests and the cat
climbs up that block.
gaming.stackexchange.com/users/48305/Jack
another way would be to distract the cats by putting another chest somewhere nearby.however, the cats will not change chests so put these decoys down before they mess up your chest opening abilitiesif all else fails, a sword is the answer
gaming.stackexchange.com/users/34312/Peter Jessen
Cats will now sit on chests, beds, and furnaces until you give them a fish.  While a cat is sitting on a chest, the chest cannot be opened.  Is there any way I can prevent a cat from sitting on a chest while still being able to open the chest?
How can I prevent my chest from being sat on?
gaming.stackexchange.com
<minecraft-java-edition>
gaming.stackexchange.com/questions/58294
gaming.stackexchange.com/users/None/
For liquids there is water and lava, for plants there are leaves and cactus. For blocks there is glass, slabs, stairs, monster spawner, snow and ice. You can also stack chests on top of chests for more storage. The problem can be solved easily by using in game mechanics.
The community suggests to use certain type of blocks or get rid of any animals within the chests range.