Backpacks, there have been quite a few mods out there that accomplish this feat, but what if you don't want to open forge every time you want to play Minecraft, or have a server but don't want to make your friends download the mod as well. Well here is the solution... completely in Vanilla Minecraft!
~~ Installation Instructions ~~
1) Download the .zip file, this contains both the Resourcepack and Datapack
2) Drag and drop the ResourcePack to your ResourcePacks folder (%appdata%/.minecraft/resourcepacks) Make sure you enable the ResourcePack in game!
3) Create a new world that you want Backpacks to be in (You can skip this step if you already have a world that you want backpacks to be in) - but make sure cheats are enabled!
4) Drag and drop the Datapack to your worlds save file under the datapacks folder (%appdata%/.minecraft/saves/<Your world name>/datapacks)
5) Open your world and type /reload! (Cheats must be enabled)
Warning! If you do not see the crafting recipes as an image in the chat when you first launch backpacks, the ResourcePack was not properly installed! Double check to make sure you have correctly installed it!
Recipes (To be used in a normal Minecraft crafting table):
X = Leather Z = Enderchest
Y= Leather Chestplate D = Diamond
B = Chest G = Golden Chestplate
L = Dropper H = String
J = Item Frame A = Armor Stand
O = Chain
Large Backpack:
XYX
B
Medium Backpack:
XLX
Small Backpack:
HJH
Ender Backpack:
DGD
Z
Armor Backpack:
OAO