Quasiminimal sets are sets for which a pertubation can decrease the area but only in a controlled manner. We prove that in dimensions $2$ and $3$, such sets separate a locally finite family of local John domains. Reciprocally, we show that this property is sufficient for quasiminimality in every dimension. In addition, we show that quasiminimal sets locally separate the space in two components, except at isolated points in $\mathbf{R}^2$ or out a of subset of dimension strictly less than $N-1$ in $\mathbf{R}^N$.