White City House, London: The private club and hotel for fans of Doctor Who, Dad's Army and Fawlty Towers

For anyone who grew up watching Doctor Who, Dad's Army and Fawlty Towers, a new hotel in London offers an intriguing chance to peek behind the scenes where they were made.

White City House is in BBC Television's former London headquarters – a Grade II listed building with a distinctive circular core that has itself made several cameo appearances in BBC shows.

The property is the latest instalment in Soho House's global empire of private members' clubs. In less than 25 years the group has blossomed from a single "house" in London's Soho to a sprawling network of 23 properties, mostly in Europe and North America. Members are largely from the creative industries (think advertising, music and media) and the atmosphere is emphatically informal – the antithesis of the jacket-and-tie stuffiness of a traditional gentlemen's club. By definition they're exclusive, so for non-members there's a tantalising, I-wonder-what-goes-on-in-there mystique.What's not well-known is that many houses also have hotels that can be booked by the public. Guests get temporary club membership and can use all the facilities – a savvy marketing strategy that no doubt encourages many to join.The club's 45 bedrooms range in size from "cosy" (15 square metres) to "medium" (35 square metres) and are surprisingly affordable with rates starting at £100. In homage to the building's '60s heritage, the decor is fiercely retro-chic, with dark wood panelling, mid-century furnishings and black-and-white tiled bathrooms. Even the technology is vintage, with a Roberts radio and an old-fashioned rotary dial telephone. Thankfully, the emperor-size beds are modern and comfortable with luxurious Egyptian cotton sheets.

The period styling won't be to everyone's taste, but the real reason you stay here is to gain access to the club. After flashing my temporary membership card, I pass BBC-inspired artwork on my way to the communal lounge on the ninth floor. It's a sprawling, open-plan space with a games room, two bars, an Asian-influenced restaurant and a plant-filled terrace with sweeping city skyline views. Again, the styling is mid-century with dark woods, vintage light fittings and muted colours. Downstairs is a studio for the club's busy schedule of member's events (there's a restaurant takeover the night I stay) and upstairs on the rooftop is a cabana-lined outdoor pool. There's a second pool in the basement along with a sauna, steam room and by far the largest gym I've ever set foot in – an enormous wood-panelled cavern that was previously the BBC's archives.

If you need to work, there are designated laptop spaces but the entire club becomes laptop-free after 7pm. Other rules include no phone calls, no posting on social media and a "relaxed dress code". Over the course of the day the club transitions seamlessly from convivial workspace to buzzy restaurant to party spot. There's live music every night and a DJ to the wee hours.

Step outside and you're in an area that many are claiming is "the new Shoreditch". Located just north of Shepherd's Bush, White City was once an uninspiring, forlorn pocket of West London. Now it's a development hotspot, home to Europe's largest shopping centre (the gargantuan Westfield London), which in turn has lured property developers and restaurateurs.

As someone who was weaned on a diet of BBC television, staying in its former headquarters has a particularly nostalgic appeal. But even if the building means nothing to you, White City House still provides an intriguing glimpse into an exclusive and otherwise inaccessible world.

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