As undrentide-san explained, わけ(が)ない denies the whole preceding clauses うちが住んでるのに田舎(だ). In this case, ~のに "even though" performs the same function as ~んだから "because" as a result. See the following example.
地面が濡れてるのに晴れてるわけがない。 (= 地面が濡れてるんだから晴れてるわけがない。)
There is no way that "even though the ground is wet, it's fine". (= Because the ground is wet, there is no way it's fine.)
うちが住んでるのに田舎なわけない。 (= うちが住んでるんだから(ここが)田舎なわけない。)
There is no way that "even though I live here, this town is rural". (= Because I live here, there is no way this town is rural.)
Incidentally, うち can refer to "we/my family" in Standard Japanese. These two うち have different pitch accents.