Sometimes you can, banking sites are very secure and tend not to have a way to remember passwords, Hotmail, MSN etc do.
Logging into a machine isn't that hard, but if the person who's repairing your machine does it for a living, why would they be interested in signing into your Hotmail account ?
Signing out on a different machine wouldn't sign you out on the original one, but if the password isn't held, it wouldn't let that person sign back in, and most sites have a cut-off if you're not logged in for a length of time (the machine dying tends to log you out).
Sometimes people forget passwords, or they get hacked and changed on a different machine. Unless that person knows some very specific data about you getting the password reset from MSN (as obviously the new one isn't on your machine) is going to be very difficult, again, it can be done, but for 99.99999% of people, nobody wants to.
If you're worried, use the one in library to log in, and change your password, then log out. Whatever is stored on your old machine won't matter now, even if you had it set to auto log on, as the password held would be wrong.