In general, the étale topology does not form a topology in the strict sense. However, is there any subcategory of $Sch$ where we can realize the étale topology as an honest topology on some scheme?

I think Senor Borger speaks truth. Maybe I am talking nonsense, but maybe this could be a sketch of some way to proceed (maybe not....): Key feature of a site which we cannot have in a classical topology is that there may be several distinct "inclusion" morphisms from a smaller open into a bigger one. Hence, we get a "fail" as soon as we encounter a scheme admitting an étale open with several inclusion maps, for example the inclusion of some open into the whole scheme. Suppose we have a scheme admitting a nontrivial étale covering (i.e. one with nontrivial étale fundamental group), this is also an étale open, but by the action of the étale fundamental group it has several nonequal inclusions into the whole scheme. So we have an effect which is impossible in a classical, "honest" as you say, topology. In the case our scheme indeed has trivial étale fundamental group, pick a Zariski open which has nontrivial étale fundamental group. Basically, take some finite covering and remove the ramification divisor, the complement is Zariski open. Now compose this Zariski open with the finite covering (which has become étale as the ramification has gone to nowhere land), same problem occurs, so again we get a "fail". Hence, to get honest topology, we need to have trivial étale fund. group for the scheme and all its Zariski opens, well.... that sounds pretty close to being just some closed points again. 

