Yes, you can own and decorate your very own house in this game! It is quite a noble goal to aspire to, so let's dig in to how you can become a homeowner.
First you will need 3,000 Rupees and 30 bundles of wood. So go sell some stuff and cut down some trees. Then talk to Bolson who is dressed in a lot of pink. Tell him you want to buy the house and he will cut you a deal of 3,000 Rupees if you provide some wood.
Now, whenever Bolson is sitting by the fire, you can talk to him to upgrade your house. The biggest use for the house is the equipment mounts. You can leave weapons here, freeing up space in your inventory. Coincidentially enough, there is enough weapon mounts for each one of the Champion's weapons if you wish to not use them.
Note: Talking to Hudson next to Bolson will trigger another sidequest, From the Ground Up, but you will not be able to continue upgrading the house until you talk to Hudson and watch him leave for the Akkala region.
The following upgrades, aside from buying the house itself, can be obtained all at the cost of 100 Rupees each (each mount is bought individually).
You can furnish the house with (1) a bed, (2) an actual door in the doorway, and (3) interior lighting.
You can landscape the outside of the house with (4) flowers around your house, (5) trees around the house, and (6) a sign that says "Link's House".
Finally, for the most practical upgrades aside from the bed, you can get (7) two weapon mounts, (8) three bow mounts, and (9) three shield mounts. It is recommended you prioritize the bed and the gear mounts first if you are strapped for cash.