On your Dashboard page, clicking the 'Unpublish' button for your ad puts an information notice at the top of the ad and removes it from the list of available sitting positions so it's no longer in the search results. It also removes the contact message box from the bottom of the ad page.
The ad isn't totally removed for a number of reasons the main one being that sitters still have notification emails in their mailboxes with links to the page. By keeping the page and adding the notification at the top, if they click the link to it, they will see that the position is no longer available.