That's not the case here, error.php is always public.
Double check your permissions and user groups also check if any SEO/SEF extension is causing this behaviour.

Finally check your error.php file for any bugs

Also,
www.melvilletenniscentre.com.au/junior-tennis2?template=system is a link to an item and
www.melvilletenniscentre.com.au/junior-tennis?template=system is a link to a category.

You should check your menu items and make sure they are not pointing to unpublished or trashed content