A typical conversation among members of my age demographic could go like this:
Person 1: Did you know that x > y?!?
Person 2: Are you sure?
Person 1: Yeah, I'm sure.
Person 2: Are you sure sure?
Is this grammatically correct?
I don't think "grammatically correct" is really a meaningful issue in respect of this relatively common spoken usage - but if I have to have an opinion, I'd say it's valid but informal.
In general, to be a [noun] noun], or be [adjective] [adjective] is simply an informal way of adding emphasis.
Often it's because the word being repeated has acquired multiple shades of meaning - arguably in OP's example the word sure can mean anything from "Okay, I guess so" to "I would stake my life on it". The repetition is intended to focus attention on the primary meaning (certainty).
Modern slang in North America (it doesn't seem to be restricted to a single age group) doubles a word to indicate emphasis, or in some cases that it's "really" whatever the word is - either that it truly literally is, that it is some sort of examplar of the word, or that it is a lot of whatever the word indicates.
Language Log has quite a few mentions of this. Try http://languagelog.ldc.upenn.edu/nll/?p=3286 , http://itre.cis.upenn.edu/~myl/languagelog/archives/004591.html and http://itre.cis.upenn.edu/~myl/languagelog/archives/004594.html for starters - they have links to research papers, and/or comics. Can't lose, really.
I agree with both FumbleFingers and Kate Gregory.
However, in this particular case, it seems to me that the sentence actually implies: "Are you sure [that you are] sure?". If the dropped words are correctly understood, the sentence makes perfect sense. It is not merely adding emphasis.
Grammar may not allow dropping words arbitrarily, though.
It is correct in that it is understood and a recognizable part of speech. I would consider it slang but whether or not slang is correct, that's a question for endless debate. I do disagree with the idea that this emphasis is only used on nouns as one of the most common use cases I am aware of is for emphasizing a verb.
Yeah, I like Susie but I don't like like her.
I'm not sure of the correct way this is written, probably because it is rarely written. (like like, like-like, like like, etc...)