There are two possible sets of answers to your questions, depending on your approach to language use:
1. No and yes!
No, it's not grammatically correct, and yes, if someone says this they mean 'I
didn't do anything.'
2. Yes and no!
Yes, it could be seen as 'correct', as millions of native speakers in the USA, Britain and elsewhere say this every day. If rules of language are descriptive, then this is a 'correct usage' = it's consistently and widely used, without ambiguity, within native speaker communities.
Some 'grammar nerds' will tell you that in grammar, as in mathematics, two negatives make a positive, and therefore if someone says 'I didn't do nothing' they actually mean 'I did something'.
Personally, my answer would be 'It depends and yes.'