Cantor illustrated the concept of infinity for his students by telling them that there was once a man who had a hotel with an infinite number of rooms, and the hotel was fully occupied. Then one more guest arrived. So the owner moved the guest in room number 1 into room number 2; the guest in room number 2 into number 3; the guest in 3 into room 4, and so on. In that way room number 1 become vacant for the new guest.And here's a more prosaic, but entertaining, take on Sam Cantori and his hotel.
What delights me about this story is that everyone involved, the guests and the owner, accept it as perfectly natural to carry out an infinite number of operations so that one guest can have peace and quiet in a room of his own. That is a great tribute to solitude.
Update: as commenter hfv points out, I (and Peter Høeg) got the attribution wrong. It's really Hilbert's story.