You lose 1 percentage point of health per day if you are homeless. If your health reaches 0, you will die and lose all your assets immediately. You are able to recover your character after the death, but you will not be able to reclaim your possessions.
You can rent your own home from an Apartment Renting company, via the World Map. Look for pieces of land with a home icon on them, indicating that an apartment building has been constructed. You will pay a daily rent when renting an apartment.
Note that only Apartment Renting companies may rent out apartments. They must own a piece of land with an apartment building constructed on them, and then they must define the number of available apartments for renting and the default daily rent via their Company Management page.
You can recover lost health by consuming Medicine Boxes, purchasable from retail stores. Read more about retail stores in the game manual article Energy & Retail Stores.