You have to leave at least one email address in your FaceBook settings, for contact purposes with FaceBook itself; however, you can control who, other than FaceBook, can see that email address.
Go to your FaceBook page in a browser (desktop would be best).
Click on your name at the top of the left column, so that you can see your Profile.
Click on Update, bottom and towards the right in you cover pictures.
Click on Edit in your Contact information.
At the top of the list are all the email addresses you've given FaceBook. There are two sets of controls. The one on the left determines who can see that email address. The one to the right of it blocks the email from showing up in your timeline. For complete lock down on that email change it to Only Me and Blocked.
When you follow the instructions in The Ob's link to delete email addresses, make sure the "Allow my friends to include. . ." check box is not checked.