Do you hire a car or you rent a car?

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Iza hires my car bossman.

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Here in the UK it’s been traditionally known as car hire. 

But the American term car rental has also become very common indeed and is now used interchangeably. 

I probably habitually type the search term “car rental” these days whenever I wish to hire a car. 

Car hire is a term which is slowly becoming more associated with getting a chauffeured car service. Such as wedding or funeral car hire. Or those bloody awful stretch limo hire services which usually smell inside of spilt stale alcohol, puke, and cheap perfume...

You are too overcoming it.

You buy a car.......

It depends on the activity.

If you are going to a care rental agency, like Hertz, Avis, Dollar etc., and no driver comes with the vehicle because you are the driver, then you are renting the car.

If you called a taxi, cab or "livery service", then you are hiring a care.

If you are using something like Uber, Lyft or any of those "not a cab, but a person is paid to pick you up and drop you off for a fee" companies, they are ride sharing (even though sharing has nothing to do with the business model), then that would be "ride sharing", "ride hailing" or any of a number of other "not a cab, but a person is paid to pick you up and drop you off for a fee" companies, then it seems you can call it whatever they want you to call it, except not a cab (or taxi) company.

I would use rent a car.

Usually hiring something would involve the owner of it operating it.

Or you hire someone to clean your house.

Consider the word's use in "renting" a flat rather than "hiring" a flat. (Apartment to non-English speakers)

I do see that the term "Hire Car" is becoming more prevalent though, but it is an Americanism.