ある is a verb which means "to be located" (It also means 'to have', but that is not relevant here).
The negative conjugation of ある is ない. Which behaves grammatically like an i-adjective. Whether it is strictly correct to call it in i-adjective, I do not know.
So ない means 'is not located'. Therefore your sentence can translate as "It is located neither near nor far from here".
じゃない means 'is not'. But it denotes the lack of equality of two things. e.g ."A cat is not a vegetable". 猫は野菜じゃない. Can you see how this is different from the "is not" in "He is not here"? In the latter we are talking about a location and not about the equivalence between "him" and "here".
Note, that I am confident about the things above this line, but less so about the things below.
Unfortunately things can get more complicated. If you wanted to say "It is not here" you could say ここにない but you could also say ここじゃない. I'm starting to speculate now, but I think the latter could just be an abbreviation of あるところはここじゃない. That is to say that the equivalence we are comparing is, the place where it is located, and here. Which makes more sense than comparing it (which maybe a cabbage) and here, which is a place.
For this reason, as a non-native speaker, I'm not sure if 近くも遠くもじゃない is grammatical. It sounds weird to me though.