Which of the two following sentences correctly uses "ours"? Is it correct to write, "Why fight a war that isn't ours?" or "Why fight a war that isn't our's?"
The personal possesives pronouns our and ours--the possessive form variations of the personal plural pronoun we--never use an apostrophe to indicate possessiveness. The reason is because of the word class they belong to. The word class they belong to is that of personal possessive pronouns. No personal possessive pronoun in English takes an apostrophe to indicate its possessiveness. This is because the entire word is the possessive form. There is no instance in which the word our or ours is not a possessive. Since this is true, there is no need for the redundancy of the possessive indicator 's. Other personal possessive pronouns are his, its, whose, their, hers, ours, yours, theirs. None of these personal possessive pronouns take the possessive marker 's, because each entire word is the possessive form and never used in any other way.
Additionally, when personal pronouns are written with apostrophes, they are contractions of the pronoun plus a to be or a to have word: it's (it + is; it + has, as in it has got: it's got), they're (they + are), who's (who + is; who + has as in who has got: who's got).
In contrast, indefinite pronouns like anybody, somebody, anyone, everybody, everyone, nobody, no one, one, someone, do require the possessive marker to make them possessive. These words are not personal possessive pronouns; they are indefinite pronouns for plural or singular entities. To indicate possession associated with this class of word, the possessive marker is required: anybody's, somebody's, anyone's, everybody's, everyone's, nobody's, no one's, one's, someone's.
The correct way to write this sentence is the option that does not have an apostrophe in the word "ours." So you would properly say "Why fight a war that isn't ours."
In general, you would think that there would or should be an apostrophe there. Apostrophes are generally used to show that a word is in the possessive case. Sadly, English is not a very rational language and there are all sorts of exceptions.
In this case, the apostrophe does not get used and you say "ours" not "our's." Please follow the link for more discussion of this and for other places where apostrophes are and are not used.