Stolen from Richard Elwes:
The best “real world” example of Russell’s paradox I’ve seen (I forget where now) is this: A librarian, bored with work, sets about making two indexes of books in their library. In book one, she lists all books which don’t reference themselves. In book two, she lists all books which do reference themselves. But then she thinks: I’ve just created two new books, so I should list them as well. But neither book references itself, so they should both be listed in book one. But by doing this, book one now does reference itself! So book one should really be listed in book two instead. But if she does this, then book one goes back to not referencing itself!
I like it. Much better than the barber one.