"Empty Space" is a name that is a little misleading from the standpoint of modern physics.
Suppose we evacuate a vessel, and clad it in impenetrable radiation shielding, so that there are no molecules or atoms within, and there are no gammas or cosmic rays. In everyday speech, we would say that the vessel has "empty space within it".
However, modern physics thinks of space and time to be made of quantum fields: a handful of them. So, our everyday statement would more accurately be rendered: in this region of space, the electron field is in its ground state, the electromagnetic field is in its ground state, and so forth. The quantum fields are in their ground states, but they most assuredly are still there. The process of the vessel's evacuation can now be thought more accurately as the driving of the quantum fields into their ground states, rather than an "emptying process".
Likewise for the question about the "edge of the Universe". We can only exist in spacetime that is already there. We can't reach an "edge" and "make" spacetime as we cross it. It is quite possible that the Universe is topologically compact, meaning roughly that one can in principle follow a "constant direction" path (this notion has some fiddly technicalities, but think of following a great circle on a spherical balloon) and not keep going forever, but rather return to one's beginning point. There are no edges to such a spacetime. Moreover, there is not an "outside" to the Universe: such a thing simply does not exist by definition of the "Universe". It is possible that the universe is finite and compact without edges, or it could indeed be infinite. If the former, there is still no "outside".
So "empty space" really is a kind of "stuff" that can have inhomogeneous properties: in General Relativity, one can quite definitely say that this piece of spacetime over here can have different geometrical properties from another piece over there.
In answer to your title: we, just like empty space, are made of quantum fields, so we couldn't exist without there at least being the possibility of quantum fields in their ground state, i.e. the existence of empty space is a necessary condition for our existence.