To start this quest you need to talk with Earl, a hunter in the bar near the south entrance of Junktown. You will need to have a few caps with you to buy him a beer via dialogue (already having a beer with you is not good enough - he wants the bar to get a sale). 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 Old 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. People have had trouble once they get Wally's name, and go to him immediately, and he only says "Ugh" or something, but that means they need to talk to more Necropolites who tell them where to find Wally. It doesn't matter if you already know where Wally is - you need to talk to people about where to find him to get the proper dialogue options.