The apostrophe can be used for the contraction of "it is", but also to show possession such as "Jimmy's bat", in this case "It's old fence". The intention is not to show the plural as in "its all up to you". I think the "it's" form is correct.
Its is a possessive pronoun meaning, more or less, of it or belonging to it.
And just for the record there is absolutely, positively, no such word as its'