Not for a little while, once you get to Novac and they like you enough you get a crappy hotel room. If you continue with the main quest then you can get to the Strip and get several suites in different hotels that can be upgraded for fairly cheap.
I just stored my stuff in a vending machine at Goodsprings for a while.
/e: since this is in other games you probably have the PC version, there are also several mods to add housing.
This post was edited by hellwar on Nov 14 2010 10:10am