Houses are offered to you by the Jarls of Whiterun, Riften, Markarth, Solitude and Windhelm. In all cases, there are things you must do before you are allowed to buy the house, so you should speak to the Jarl and see what he or she needs you to do, as well as talking to their stewards for the same. Eventually the offer of thaneship will come up, this will typically open the opportunity to actually buy the house.
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There are many houses in Skyrim. Each house require an amount of gold or a quest before you can buy it. I wrote a list of houses with requirements to get them. Breezehome (Whiterun Hold) you need to complete quest Bleak Falls Barrow and have at least 5000 gold Honeyside (Riften) you need to complete the quest for Jarl of Riften and 4 more jarl quests. Cost 5000 or 8000 gold Vlindrel Hall (Markarth) complete a quest "Dungeon Delving" . You need to be level 20 to get the quest. Cost: 8000 gold Hjerim (Windhelm City) quest "Blood on the Ice" Cost: 8000 or 12000 Proudspire Manor (Solitude) quests "The Man Who Cried Wolf" and "Elisif's Tribute" Cost: 25000 All houses (after quests) are bought from the housecarl of the local city jarl.
A courier will come to you with a letter from one of the holds which you can buy land from. You'll have to complete some quests/meet the qualifications required to own land.
Unless you have mods, the Hearthstone DLC is required to purchase and own land outside of major hold capitals.
You can't buy a house, you have to build one, but you need the Hearthfire add-on first.
After that, you must become Thane of Dawnstar by doing missions for the townsfolk. You must also have completed the 'Waking Nightmare' quest and a 'Kill the Giant' bounty request mission. Doing so will allow you to buy the Heljarchen Hall plot of land from the Jarl, and on this land you can build a house, if you travel to the location you'll get all the instructions there.
There are several houses you can buy in Skyrim. The first and least expensive is the "Breezehome" in White run. It can be purchased from the Jarls adviser in Dragonsreach for 4000 gold.
kill the first dragon and ask the guy who sits next to the jarl in dragonreach, by the way its 5000 gold.
Yes you can buy a house. To buy a house in a city you first need to be Thane of that city. You can then talk to a city steward and purchase a house, provided you have enough gold.