There are many very entertaining discussions of "Hilbert's Hotel" online. I look them up once or twice a year for teaching my math classes. To answer your very pertinent question, I suggest looking at
for an illustrated, and I think illuminating, discussion of what to do if countably infinite buses, each with countably infinite passengers, drive up to Hilbert's Hotel with infinite rooms, each of which is already full.
http://opinionator.blogs.nytimes.com/2010/05/09/the-hilbert-...
for an illustrated, and I think illuminating, discussion of what to do if countably infinite buses, each with countably infinite passengers, drive up to Hilbert's Hotel with infinite rooms, each of which is already full.