Upon entering Vault 11, the first thing to note is four skeletons at the entrance door and a terminal nearby. Accessing the terminal will download a recorded holotape and a note to your Pip-Boy, inform you to determine what happened to inhabitants of the vault. After picking up the holotape, head into Vault 11. There are many terminal entries throughout the vault's levels that contain hints as to what happened to the residents of Vault 11; however, you don't have to find any of them except for the overseer's password. To obtain the password, travel to the Vault's atrium.
The atrium is located past the overseer's room and downstairs. On the podium there is a holotape labeled "Prepared Speech of Gus Olson, ombudsman, for the Annual Overseer Election"; pick it up. Alternatively, you can download the holotape upstairs on a terminal. Once the speech is read, you will find out that the password for the overseer's terminal is the first name of the first overseer's wife, Betty. Now, go back to the overseer's room and unlock the sacrificial chamber.
Follow the hallway and watch the video. Be prepared for a challenging battle; there will be a sentry bot, Mister Gutsy, protectron, robobrain, and several turrets waiting to attack once the video is finished. Kill all of the robots and open the only other door in the room. Finally, download the two holotapes on the mainframe and you will be rewarded 500 experience points.
If you have chosen the Robotics Expert perk then you can simply use a Stealth Boy and deactivate all twelve robots and then use the mainframe to unlock the door back out to the overseer's office.
Throw some pulse grenades at the side walls (before they open), and they will shock some of the robots. If you have pulse mines, they can be tossed through the walls before the doors begin to open (they actually seem to pass through an outer wall and be stopped by an inner door, but the robots will still set them off when the inner door opens and they emerge).
The mantises and giant rats may set off traps before the player ever reaches them.
On un-patched versions of the game there is a repeatable 500 XP exploit on the final terminal in the sacrificial chamber. See the Fallout: New Vegas exploits page for details.
Behind the scenes
The pre-execution film and audio in the sacrifice room could be a reference to the assisted suicide sequence in the 1973 science-fiction film Soylent Green.
The music that was played during the walk into the sitting room to watch the film is the same music that appears during the ending film in Fallout 3 and Fallout: New Vegas.