Can calibre connect to the internet? e.g. does Metadata download, Get plugins, Fetch news, Get books work?
If any of those features work, something is probably blocking email specifically. Go to Preferences ==> Sharing books by email and check your emil settings then click "Test Email".
If it fails (sounds like it will) then post the contents of the log. (Make sure to redact your username and password!
)
If calibre is being generally blocked... do any other programs suffer lack of internet connectivity?