"(그 사람들은) 이렇게 안 있지 않을까?" ('그 사람들은' is added as an example to help explain it) => Wouldn't (they) not stay/be like this? == (They) would not stay/be like this. The important thing to understand is how we ask the negative in a suggestive form to make a point of the positive, and sometimes vice versa. That is, 않을까?, 않는가?, 않아?, 않니?, etc. at the end of a yes-no question (i.e. with no question word like 무엇, 왜, 언제, etc) is very often rhetorical, not asking a question but making a point instead. This is especially true with the subjunctive-like forms like 않을까? and 않는가?, and in monologues, but is not limited to them. It also happens with a rhetorical positive questions as in -겠어?, -ㄹ까?, etc. For example: - 이 옷 예쁘지 않니? (= 이 옷 예쁘다. 그렇지?) Isn't this suit pretty? (It is pretty, isn't it?) - 멀리 그가 오는 것이 보이지 않는가? (= 멀리 그가 오는 것이 보인다) = Don't I see him coming up ahead? (= I see him coming ...) Your example is more tricky because the main part also has the negating 안. Even so, you can still remove the negator 않을까? at the end to construct the meaning. Since 않을까? is suggestive rather than definitive, its meaning is like 이렇게 안 있겠지 or 이렇게 안 있을 거야 (they would not stay/be like this). The positive version of the question, 이렇게 있을까?, has about the same meaning as the original, as it's suggesting the opposite, 이렇게 안 있겠지 or 이렇게 안 있을 것 같아. It is clearer and easier to understand than the negative version.

Basically the double negative changes it to a positive. It can also sometimes have a stronger feel than a normal positive sentence. 이렇게 안 있지 않을까요? = 이렇게 있을까요? Hope that helped!