It's thanks to your trusty rucksack that you can carry your food rations, your water or your weapons with you. It is pretty limited though, and fills up quickly... by default it has 4 storage compartments and one compartment reserved for your uniform.
If you are in Hero mode, you have an additional storage compartment in your rucksack! In addition, you will also have to carry your citizen's uniform and your work uniform.
When you come back from the World Beyond, your House is where you can unpack your rucksack. It is also where you will fill it before setting off again.
Your personal chest
Your personal chest can hold 4 objects by default (5 for Heroes). Certain objects and home improvements allow you to increase this limit.
Certain objects, like the Carpet are automatically used to make improvements to your House if you place them in your chest. Improvements give various bonuses (decoration, protection against theft, defence against zombies...)