Vessels are likely the first means of storage the player can acquire. They can be carried and placed freely without loosing it's contents. Craft a Vessel with 6 clay balls:
Then bake it in a stove:
Once baked, you can use Right click to open the inventory. To place a Vessel hold SHIFT to Sneak, then place it with a right click.
A Vessel can currently hold the following items: Clay and Fire Clay, Food items, Ingots, Ores and Stones, Seeds, Arrows, Feathers, Dyes, Gunpowder, Flint, Leather, String, Spider Eyes, Wheat, Eggs, Ender Pearls, and Bones
Chests become available once the player has found copper. Craft with planks in a circle and one copper ingot in the middle. Apart from the different recipe they behave the same as in Vanilla Minecraft.
Tool Racks are a very useful means of storing and accessing tools. Create it within the carpenters table. Place the rack on any solid wall. Right click to place a tool and right click with an empty hand to receive it back.
Awesome looking storage of ingots. Sneak + Right click with an ingot on the ground and you will create a new pile. Add more ingots with the same method. Normal mode right click makes your character grab one ingot.
You can craft the Armor Stand from Minecraft 1.8 by replacing the stone slab with a piece of clay