Good question, but no. For example the Klein bottle or real projective plane are closed surfaces containing no boundary.

Incidentally the Mobius strip need not have a edge; you can leave out the boundary circle at the cost of having a non-compact space. This is exactly analogous to the difference between the closed and open intervals: [0, 1] vs (0, 1).

In fact the projective plane can be obtained by taking a disc and glueing its boundary to that of the Mobius strip, which makes the two boundaries "cancel".