Also "I have no car". The point is the use of singular. Besides, there is an 'ein' being used: consider "kein = nicht + ein"
This "kein" exercises are quite frustrating! "I don't have any car" cannot be considered a wrong answer... Maybe "I do not have a car" translates better what they mean in german but still... I can't understand why this is not a valid translation.
Maybe it's just me, but when I hear "I don't have any..." I expect a plural noun. So, " I don't have any car" sounds odd, while "I don't have any cars" sounds fine. Still, that would be "Ich habe keine Autos" in German. Maybe that's why your translation isn't accepts.
Would "I have no car" or "I have not a car" be acceptable here? Are they just outdated English?