Is using the possessive apostrophe correct in "the car's antenna"?

Possessor doesn't need to be a living being.

The town's hall

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Possessives in English are much more common with animate possessors, but they are not restricted to them, so no, there is no problem with "the car's antenna".

@Hamid: the possessed object is generally definite, so "the town's hall" is unusual, and implies that there is only one hall in town. "The town's concert hall" is less odd.

@The MYYN: the problem with saying "car is in the genitive case" is that that does not cover common English expressions like "The girl next door's coat". Grammatical case is a concept which barely applies in English.

In

The car's antenna is embedded in the windshield.

car is in the genitive case, here in a classifying genitive. One could also write

The antenna of the car ...

The use of the apostrophe is not dependent on the noun representing people. Using -'s is correct.

As I’ve said in answer to a related question, it’s misleading to think of the apostrophe exclusively as a possessive marker. It’s more helpful to think of it as a genitive inflection, certainly capable of expressing possession (John’s car), but also used to specify or classify the reference of a noun (the girl’s face, a bird’s nest or, indeed, the car’s antenna), to indicate time and place (a week’s holiday, the country’s capital) and to refer to a noun that is understood from the context (I’m going to a friend’s (house), Macy’s (store)).