A metaphor for recursive enumerability
I was first taken by your metaphore, but I do not think it works.
Actually he is saying that he is himself a decision procedure, though
he cannot describe his own internal program. Hence Justice Stewart
decisions are recursive, not recursively enumerable, at least where
pornography is concerned. I would indeed expect anyone capable of such
an arrogant statement to be computationally limited.

Now for other examples taken from the real world, the first that comes
to mind is actually co-RE, though that amounts to the complement being
RE.  It is the buggy program problem. If a program has a bug, it will
show some day, provided we use it long enough. But we cannot be sure
thre are no bugs, even when none has been found. Well, that was before
we could prove program correct (not that we are really there yet).

A close problem is falsifiability of scientific theories, as defined by
Karl Popper. The idea is that science is primarily co-RE:

A theory in the empirical sciences can never be proven, but it nust
be falsifiable, meaning that it can and should be scrutinised by
decisive experiments that could prove it wrong.

In other words, all we can know for certain, is that some empirical
science theories are false, because some consequences have been
experimentally proved false.
Whatever remain might be good theories, unless later proved false.

False theories form a RE set. Good theories are the co-RE complement.

But I am not sure this last example will make things clearer for your
students.

And the relations between scientific theories, or their correctness
status are a bit more subtle than that.

P.S. I think mortal people form an RE set.