Can you get your own house?

You can get your own place if you follow the main quest. For this one you can also buy upgrades.

There is an earlier opportunity to rent a motel room (forever).

There are two 'homes'. When you reach Novac, there is a Motel room you can rent, which is safe to store stuff and rest in, and if you complete the quest One for My Baby by accusing the actual perpetrator, you'll receive it permanently at no charge.

Further along the main quest, once you reach The Strip, you'll be given access to the Presidential Suite at the Lucky 38 Hotel and Casino. This functions more like the house you gained in FO3, in that when you dismiss companions, you can have them wait for you in the Suite, and you can purchase various 'upgrades' to add more style and functionality to the apartment. There are also several mods that add additional crafting resources or other customizations, if you're playing on the PC.