Francine Garret gives the Courier the key to this room when the quest Debt Collector is completed. The room contains a queen sized bed, bookshelves, and several desks. The Well Rested bonus is gained from sleeping in the bed.
Items in the room cannot be taken without being considered "stolen." The room was owned by Caleb McCaffery originally, therefore the items remain technically his, even though he is no longer staying at the Atomic Wrangler. In addition, the desks are not safe for item storage.
The Atomic Wrangler corner room appears only in Fallout: New Vegas.