In British English, "I haven't got a car" is common usage (Americans would be more likely to say "I don't have a car"). "I have no car" is also quite acceptable.
Why doesn't "I have not a car" work? It's confused by the fact that "have" is used as both as an auxiliary verb ("I have seen her") and to mean "own" or "possess" ("I have a car"). But the negative "have not", is only ever used as an auxiliary. In any case, we don't normally negate a verb by adding "not". We don't say "I walk not to work": it's clear what it means, but it's not idiomatic; we say instead "I don't walk to work".
As for the absence of an article, in the sentence "I have no car", the word "no" is essentially performing the same grammatical function as "a" in "I have a car". "a" means one, "no" means zero, you don't want to say "one" and "zero" because they contradict each other. But perhaps I'm appealing too much to logic; idioms often defy logic.
Afterthought: what about "I haven't a clue."? Not to mention "I haven't the faintest idea."? I think we just have to dismiss those as irregular.