What happens if you logon to your email id and close it instead of logoff? Can another person sharing your network have access to your email contents?

Depends on the e-mail service, If we are talking about a web based e-mail service, its possible that if they go to website, the service may note that you are still logged in and show your mailbox. But most of the time, if you close the browser, you will be logged off.

Thank you so much.. I was using Yahoo..

Actually with Yahoo, there is a little check box when you login, that asks if you would like this login to stay valid for future logins up to two weeks.

On that page, it also tells you that you can change the length of time.

So, closing the browser may or may not log you off depending on the checkbox state.