The home you can buy is in Hateno Village, southeast of Firly Pond. Cross the bridge and chat to the construction workers until you find Bolson, the head of construction.
After a bit of a chat, you'll talk him down to a bargain price – 3000 rupees and 30 bundles of wood. That's still a big outlay, but if you're fairly deep into the game you shouldn't have too much of a problem.
The easiest way to get wood is to cut down trees. Conveniently, there's a whole heap right outside Hateno. Try to use a weaker sword or axe (they'll break eventually), but if you have the Woodcutter's Axe you'll get more wood than usual. Search stables for axes if you don't have one.
3000 rupees is a tall ask, but there are various ways to collect money in Zelda: check out our easy money guide. You can also wander into town and sell some of the materials you've gathered, of course.
Head back to Bolson and give him the wood and rupees, and bam – the house is yours! This means that you'll now be able to furnish it.
To start with, all you'll have is a single weapon mount. Put a weapon up here to store it for later, freeing up some space in your inventory. You can eventually store three weapons, three bows, and three shields
Go back and talk to Bolson. He'll help you to make your house a bit nicer ... for a price. Each addition you ask for will cost 100 rupees.
Start with a bed. Once you've got a bed in there, you can sleep any time and restore your health. Yes, it's a 3100 rupee investment, but nothing beats your own bed, right?
While you're out here, have a chat to Hudson too. This will start off a new quest when he goes off to, eventually, start a whole new town. This is a fairly involved quest that will take time and effort – when you're ready for it, go and find him at Lake Akkala.
For now, though, you're a homeowner! This can become a hub for you once you've upgraded a few times – use it to skip time ahead in bed, store your best weapons, and cook food on the camp fire outside. Enjoy your little slice of Hyrule!