I believe that the の here is the same の as the の which is explained in this thread: What is the difference between the nominalizers こと and の?
Basically the の here is a noun which means "thing". It is similar to こと (noun) which also means "thing". The difference between の and こと is that の is used when the "thing" is related to the speaker. This is furthur elaborated in Derek's post: https://japanese.stackexchange.com/a/1396/264.
そんな is a pre-nominal adjective, also known as an attributive (連体詞). A normal adjective can be used in two ways, either attributively or predicatively. For example, the adjective 暑い can be used in these ways:
Attributively: 暑い日 (hot day)
Predicatively: 日は暑い (day is hot)
A pre-nominal adjective can only be used attributively, so we can say "そんな日" , but not "日はそんな"
So we can roughly dissect the sentence "そんなの答えはNOだってばよ!" into 6 parts:
pre-nominal adjective そんな: such as that, like that
noun の:
noun 答え: answer
topic-marker は (optional due to ellipsis)
quoted "NO"
copula だ
Note that since we can't simply chain nouns together (unless to form compound nouns), there exists a break between "の" and "答え". The sentence is clearer if we put a comma as such: "そんなの、答えはNOだってばよ!"
Putting it all together, a literal translation of the sentence "そんなの答えはNOだってばよ!" goes like this: "As for thing(s) such as that, the answer is/will be "NO". ってばよ!"
Putting it into the context of the story (chapter 162 or episode 92 @16:12), it means "As for a trade such as that (such as healing arms for resurrecting Tsunade's kin), the answer will be "NO". ってばよ!".
Using another example, the sentence "そんなのいらない!" can also be dissected into different parts:
pre-nominal adjective そんな: such as that, like that
noun の: thing
topic-marker は (optional due to ellipsis)
inflected verb いらない: do not need
And when the parts are merged, a literal translation of the sentence "そんなのいらない!" goes like this: "As for thing(s) such as that, (I) don't need!"
の as the no-particle
As for the の in the sentence "今のはだれ?", it is a totally different の compared to the の in "そんなの". The の in "今のはだれ?" is the genitive no-particle. Basically the function of the no-particle is to convert a noun into an adjective so that it can describe (modify) a noun. This is elaborated in this article: Genitive Case.
The sentence "今のはだれ?" is dissected into:
noun 今: just now
genitive particle の (converts the noun 今 into an adjective to describe 人)
noun 人: person (optional due to ellipsis)
topic-marker は (optional due to ellipsis)
(pro)noun だれ: who
And the literal translation of the sentence "今のはだれ?" is roughly: "As for just now that (person), is who?".