No Man's Sky has received an enormous patch named the Foundation Update, which incorporates the ability to build bases. Unfortunately, like so much of Hello Games' divisive hit, it's not immediately obvious what you have to do to create your own intergalactic home. But we come with tips.
Basically, you're going to want to fly around in your ship, scanning until you get a waypoint for a Habitable Base. Once you ping one, fly to it, and interact with the computer inside. You'll be given the option to set it as your base, from which point onwards you'll be able to upgrade it in whichever way you see fit.