To start this quest you need to talk with a trapper in "Darkwaters" bar near the south entrance of Junktown. The quest does not register in your Pip-Boy so pay attention to his instructions in dialogue and all dialogue along the way. He sends you to Boneyard to find a man with a scar on his face. You'll know when you're searching in the right area when you get dialogue about it with the NPCs you ask. When you talk with the right one, he'll give you a clue which leads you to follow his path from there to Necropolis. Once again, ask around and you eventually talk to someone in Watershed who gives you his name. Then find a ghoul named Wally who is in Watershed. The bunker base costs 350k caps.
Note: If you find Wally before you've received the clues from others along the way, Wally does not give you the base-buying dialogue, so you have to talk with people in the right order to complete this quest, and with Wally last.