For the cheapest house option you'll want to visit the office of imperial commerce in the market district, where vinicia melissaeia sells the waterfront shack for 2,000 gold. Getting her disposition to 60 through some persuasion or bribes is needed first but it's worth the effort. After buying you can grab some storage upgrades from three brothers trade goods to make it more functional.

Here's something interesting though - there's actually a completely free option that's even better for starting out. If you make your way southeast from the imperial city, you'll spot deepscorn hollow near leyawiin's northeast gate. The cave just opens up when you get close, and inside you'll find everything you need - storage space, alchemy equipment, and a safe place to rest without spending a single coin.

While the waterfront shack is decent if you want to own property in the city, deep scorn hollow gives you more features right from the start without any gold investment. The location is pretty convenient too once you get used to the route.