You may build, buy or claim a chest, but the most important part is the lock. Your own chest or container may have a private lock for it which makes it available to you and whoever else you craft keys for. More intricate and advanced locks can be crafted or purchased and your chest will be locked to everyone except the most proficient of thieves.
You may buy a home in any major holding provided there is capacity. You only own a home by owning the lock, beyond that maintaining it, obeying local laws and complying with taxation are your responsibilities. You should always be aware of who has keys and potential access to your storage and domicile.
More to come — lock tiers, key crafting, the locksmith and thief trades, and the legal status of breaking in.