CarryCapacity is a mod which allows to pick up blocks and carry them in your hands and on your back.
This was made possible with the help of the awesome @Tyron providing some code samples with the official example mods.
You can pick up chests, baskets, crates, barrels, planters, vessels and anvils by sneaking and holding right click with empty hands - that includes your offhand slot! This will prevent you from sprinting, and depending on what you're carrying, slow your walk speed down further. Placing them back down again is done by sneaking and holding right click on a block.