YouTube or Vimeo would be the obvious one, I think you can make videos private on YouTube but "how private" I dont know, but that may not be appropriate.
Kaltura is good, we use it, there is a Moodle plugin for it, the commercial service is quite expensive and almost certainly not appropriate for such short term use. As it happens I was just reading a blog about cloud integration for Kaltura:
I have no idea how good those guys are or what they charge... just thought I would throw that one in.
Another service is screencast.com by techsmith, ive used that before and its ok, the players are not as good as the above options though.
To be honest could probably start a long running thread on this subject alone!