The Doom room Easter egg is one of four secret areas in RAGE referencing classic first-person shooter video games made by id Software, the others being the Quake room, Wolfenstein room, and the Developer Graffiti room. The way there is found in the Gearhead Vault, right near the entrance, but in order to unlock it the player must reach the computer office deep inside the vault and activate seven computer keyboards that will make a distinctive Doom sound.
Best to come here when there's no resistance. After the first visit there simply turn around and re-enter. Head through the parking lot, up the slope, up the next slope, straight ahead, veer left, follow the corridor and you're at the computer room.
Hit all the seven keyboards (they're the computers that with the keyboards attached), then head back to the start of the level. Just before the exit, on the left, there's a door with a flaming barrel next to it. You can now open it. Jump onto the wooden crate next to it. Duck, enter by way a Doom style pressure plate.