The Shoggoth's Lair is located beneath the Orphanage. There're some pretty neat things down here!
To get to the Shoggoth's Lair you must go to the basement of the orphanage, make sure the thermostat is set to [REDACTED] so that the furnace is turned off and then head into it, go down and explore the cave complex, look for an altar. To the north, south and east of the altar are calendar stones, [REDACTED], then head back to the altar and jump into the pit. Don't worry, you won't get hurt... most likely. It will be dark, but if you look on your map and look for ( ) on the northwest side of it, that is the Shoggoth's Lair. You'll most likely be under the influence of fear while you're here. Just head there for either the mutation Fibrocartilage, or for the Baby Farmer job. The exit is all the way to the south.