MAYOR BARRY says he does a lot of valuable thinking and working when he is being driven around in his official car. So the mayor plans to keep the car and three chauffeurs, although he has directed that six chauffeured cars for city workers be taken out of action as a budget cut. But wherever Mr. Barry did the thinking that led him to keep his own car while doing away with all others, it was a political and public relations calamity.

Mr. Barry must realize that chauffeured cars for public officials represent a perennially hot public issue and a privilege of serving in government that drives the voters nuts. He could have picked up pointers on this from President Carter, who received great public huzzahs when, at the outset of his term, he cut the use of limousines for some officials -- even though it added up to only a minor budget saving. And the mayor could have remembered his own limousine image when he rode through a snowstorm that stalled much of the rest of the city.

Mr. Barry's biggest mistake, though, was surely simultaneously to take away the cars that serve the other city officials and council members while he was keeping his own. The move couldn't have been better designed to stir resentment and ridicule. Our own view is his giving up the car would be pointless and probably harmful; the mayor should get around town freely. And savings from eliminating his car would be tiny, at best symbolic. Other economizing steps are available to him -- steps that could entail some mayoral sacrifice. So Mr. Barry did not need to get into this bind in the first place. But he's in it now. How he gets out, we leave to him. Our only advice is this: he should do his thinking about the problem somewhere other than the back seat of that car.