Why is it incorrect to say "Without a car, it takes a long time to get from a place to another."?

Without a car, it takes a long time to get from a/one place to another.

A place = one general example of a place taken from a non-finite number of places.

[The place = that place of which we (speaker and listener) are both aware.]

One place = one specific example of place taken from a limited number of places: One place where you will find apples is below an apple tree.

Another is a pronoun with its referent as a previously mentioned substantive.

The statement also implies that it is normal to drive a car on a specific journey and so the places are specific: it is journeys like these that are referred to.

1 Without a car, it takes a long time to get from a place to another. = Without a car, it takes a long time to get from any one of an infinite number of places to another place from among an infinite number of other places

2 Without a car, it takes a long time to get from one place to another. = Without a car, it takes a long time to get from a specific place to another specific place.