I’ve been wondering why it is “me” instead of “I” and it is “being” instead of “be”?

I don't know why you think "being" should be "be" - but it is "being" because it can be a participial phrase. Basically, "being" is a participle (a verb that we add -ed or -ing to, so it becomes an adjective to describe a noun) that works as an adjective to describe the noun "me."
https://owl.purdue.edu/owl/general_writing/mechanics/gerunds_participles_and_infinitives/participles.html
https://www.grammarly.com/blog/participial-phrases/

Another identical grammar structure to this is a gerund phrase. A gerund is a verb that we add -ing to the end of, so that it becomes a noun. When you use a gerund phrase, you can say "MY being able to buy a car at that age..." because "being" is a noun and "my" is your possessive article attached to it.

Technically, this sentence is "wrong" both ways. If it were a proper, correct participial phrase, we would write "I, being able to buy a car at that age, was an exception." This is correct, but it sounds really stuffy, and if you type it without the commas (I being able to buy a car at that age) looks completely incorrect.

The same goes for "my" - you will see the words "my being..." quite often in English, but it's very uncommon in speech. It's still more common than "I" in my opinion, though. But most often you'll see the words "me being" because these phrases often come at the end of a sentence. (Ex. That was just her being rude. They were annoyed at us being loud. My mom doesn't like me being upset. We want to see them smiling. etc. etc. etc.) "Me being" is more common, so "my being" - even though it's grammatically correct - starts to sound more strange.

So, in common speech, we like to fuck up these phrases by using the object pronoun + gerund/participle, no matter if it's the subject or object :)

I know the first part of the above sentence is the subject, so it should be nominal.

If it was a correct participial phrase - yes, we would write "I, being able to buy a car at that age, was an exception..."
This also sounds a bit wrong though, because you need to section off "being able to buy a car at that age," and that makes the sentence slightly different. The original one makes the focus on being able to buy a car. THAT is what is exceptional. When you "correct" it to use the word I, and then you section off the participial phrase in commas, it makes it sound more like you're saying "*I* was exceptional," which sounds a little more arrogant than intended.

What I was able to…?

"What" is mostly only for questions

That I was able to…?

You could say: That I was able to buy a car at that age was an exception.
Same exact meaning as "Me being able to buy a car at that age was an exception."

“Me being able to” seems a very new structure to me, which I don’t think I see often.

I don't know what dialect you prefer, but just letting you know my observation on the phrase "seems a very new". This is 100% correct, but I associate it with British English. In American English, I would prefer to say "seems to be a very new" or "seems like a very new".