As in many languages, negative sentences in English select certain words (called "negative polarity items") and disallow others. A particular set of negative polarity terms is the "any" series: in negative sentences and questions, "some" words are generally replaced by "any" words, eg
I found somebody. vs I didn't find anybody.
In many dialects of English, the negative polarity terms are "no" words instead of (or as well as) "any" words, so:
I didn't find nobody.
These forms are not used in any Standard Englishes, and so the idea has grown up in the last couple of hundred years that they are "wrong". As is often the case with forms which some people use but which the establishment judges to be "wrong", a spurious rationalisation has been invented for that judgment (rather than admit that it is an arbitrary rule): that "I didn't find nobody" means "I found somebody". This rationalisation has its intellectual base in logic (but much language is not logical) - that it is false can be seen by the fact that even English speakers who do not use such forms themselves always understand them (unless they are being deliberately perverse): thus both the speaker and the hearer understand "I didn't find nobody" to mean "I didn't find anybody", and it is false to claim that it "means" "I found somebody".
However, while it is not true in general that "a double negative means an affirmative", it can do, given suitable context and prosody. Stephen Pinker gives the example "Try as I might, I must admit that I can't get no satisfaction from this", where it clearly means "I do get some satisfaction".
Your example is another case where the negatives do cancel out, and I think the reason is that the order of negatives does not allow for another interpretation. If I heard "He hasn't never been in the family", I would take that (special prosody or context aside) as a non-standard form of "He hasn't ever been in the family", with the non-standard negative polarity term "never" replacing the standard "ever". But what we have here is "he's never not been in the family", with "never" preceding "not". That is not a position where a negative polarity term is expected, so it is not a non-standard version of anything. Thus it can only be interpreted as a negative in the scope of a negative: it is grammatical, and both have independent meanings.
Note that they do not just cancel out: it doesn't mean "He has been in the family": it means something like "He has always been in the family", but with an added connotation, something like "contrary to what has been suggested".