|This needs a stretch. (Feel to remove when satisfied of completion.)|
Although its location is kept hidden from the public (even from Mystery Inc.) it's likely in or around Gotham City, the city the Dynamic Duo have sworn to protect. The entrance is concealed and blocked off by a wooden gate with a "Road Closed" sign (which can is designed to come forward onto the ground, instead of move sideways), surrounded by bushes.
It has all types of sophisticated tracking equipment, and is also where their vehicles (such as the Batmobile) are stored.
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