It depends on what you mean by an étalé "space". As long as $C$ has a small set of topological generators (i.e. as long as $Sh(C,J)$ isn't too large to be a topos), there always exists a certain version of an étalé space: If $F$ is a sheaf on $(C,J),$ the slice topos $Sh(C,J)/F$ has a canonical étale projection $$\pi_F:Sh(C,J)/F \to Sh(C,J).$$ This map is a local homeomorphism of topoi. This topos with this local homeomorphism is the étalé space of $F.$ Indeed, we may make this construction for each object $c \in C,$ call it $U(c):=Sh(C,J)/y(c),$ where $y(c)$ is the (possibly sheafified) Yoneda embedded object. Then, "sections of $\pi_F$ over $U(c) \to Sh(C,J)$" are in bijection with elements of $F(c).$ If the Grothendieck site $(C,J)$ happened to be the canonical site of a topological space, then each slice $Sh(C,J)/F$ is equivalent to sheaves on the étalé space of that sheaf, and the projection corresponds to the usual one. In particular, $U(c) \to Sh(C,J)$ corresponds to the inclusion of an open subset. So, this is reduces to the usual construction for spaces. Another example is, if $(C,J)$ were the small étale site of some scheme $S$, then each $Sh(C,J)/F$ is the small étale site of some algebraic space (with no seperation conditions) étale over $S,$ with $\pi_F$ corresponding to the étale map from this algebraic space to $S.$