Still trying to find pictures of my St Joe Folly House, I bought a little book assembled by the Museum Hill Historical Society. It doesn’t include THE house, but it does include this one with a tantalizingly similar secret tunnel.

Transcribing: An enormous secret tunnel that once led to the Nunning Brewery can still be accessed through the basement. Nunning had a bar in the basement with taps fed from the vats in his brewery.
Incidentally, the Nunning house was for sale recently, priced at $650k, with a fantastic set of pictures here.
Definitely not the same house. My memory of the outside is vague. The main part was a 1920ish bungalow with Mission-style stucco exterior, maybe like this Radford plan:

It had a 1940ish two-story wing on the right with several small apartments. Off to the left was the special part, which I remember more clearly:
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Below the main wing was a basement containing a museum, with archeological displays and skeletons. From the basement a long tunnel, something like the Hartness tunnel, led down to a large underground auditorium carved from a natural cave. (I’m calling it an auditorium because it had a raised stage at one end.) The auditorium had a separate hillside entrance, something like a mine adit, with a long flight of stairs down to the auditorium. This entrance, and the auditorium and the apts, were built like a school or public facility, with industrial-type floors and walls. There was a little ticket booth inside the hillside entrance.
What was it? I haven’t the slightest idea. It might have served as a bomb shelter, but the school-type floors and stairs were older than the ’50s, so weren’t built for that purpose. The tunnel had shelves with full wine bottles, and the attic above the apts contained old pinball machines and similar gambling equipment. Best guess: a speakeasy or semi-legal private nightclub for guests of the manor? Gambling + wine + secret room with a stage = speakeasy? Maybe.
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More rehash for completeness: I visited there in 1960 when my uncle was renting the main house. He was serving as manager of the apts. The house was on the west side of a north-south street. There was a school just downhill (west) from the house, and the school playground equipment was between the school and the house. I think the school was unused at that time. Just to the right of the apartment wing, a long driveway ran downhill to a multi-car garage for the apts. The hallway of the apts had a door on the driveway, and a door into the living room of the main house. I’ve never found the location on Googlemaps, but 14th at Charles seems to fill the bill in most ways.
