Why is Dr. Who’s time-traveler ship, the Tardis, shaped like a British police box? Jill Lepore answers in a New Yorker piece marking the fiftieth anniversary of the TV show’s first episode. In fact, there were several answers.
One was faux-scientific: “The outside appearance of the machine is a police box because when the machine is made and before it goes critical it is given an anchor in a definite age and space, without which there can be neither past nor future, and the time/space traveler would go mad—or meet God.”
The more interesting one is that “the Tardis is supposed to change its shape to blend in with the local surroundings but that its chameleon circuit is broken, so it’s stuck being a police box.”
Probably the real answer is that the show’s “production budget could not possibly accommodate building a new ship for every adventure.”
Whether intentionally or no, the police box fits: “British police are called bobbies because the London Metropolitan Police, a model for police forces all over the world, was created by Home Secretary Sir Robert (Bobby) Peel, in 1829. Doctor Who polices worlds. The idea of a world’s policeman dates to the First World War and began to come into common usage near the end of the Second.”