Here goes nothing....
First this may not be an answer, its just a chain of thought that might help isolate the actual issue, by the looks of it without admin access it would be hard to dive deeper.
First it is redirecting to 404 because it thinks the page is not there. However 404 is also not there, so it tries to redirect to the 404 page, and you can see where the redirect loop starts.
So I am guessing you have no menu item for the 404? Try to add one and point it to some article and see if that helps the redirect loop (for starters).
Then take that menu item and grab the id. After that go to http://www.melvilletenniscentre.com.au/index.php?Itemid=MENU_ITEM_ID
Does the page show up?
If so then most likely your htaccess file is not working, or you do not have the global configuration for rewrites created properly (or something with SEF). My best guess is the htaccess file, replace it with a brand new joomla installs version. For the global configuration, check the "site" tab. The settings you are looking for is Search Engine Friendly URLs
and Use URL rewriting
.
If not then it is a more serious problem, this is the baseline for how the menu manager works, so I would attempt to update Joomla to see if that helps, sometimes the newer files can fix issues (though I find that unlikely).
For the 404 page, something in your website is telling 404 pages to redirect there. There are a lot of things that can do that, but my guess is possibly the template error.php file. This file is meant to handle 404's.
if ($this->error->getCode() == '404') {
header('Location: '.JRoute::_('index.php?Itemid=156'));
exit;
}
This is an example of what you might see there, although possibly different, using Itemid like above would avoid the redirect loop, most likely yours would be going to /404 instead without using JRoute to define a menu item (this way even changing the alias will still redirect properly).
EDIT:
I have looked at that new 404 page, there is an error in your templates error.php
Fatal error: Cannot access protected property JException::$message in /home/USER/public_html/DOMAIN/templates/TEMPLATE/error.php on line 52
Take a look at line 52 there and see if there is anything that can be done there. Without seeing it I cannot really help with the exact cause but most likely you could remove that portion without much problems.
The page still reads a 404, meaning the record in question cannot be found and it has given me another thought, what is the publish dates? you can set a start publishing and and end publishing date, if they are set it could be triggering the 404. They can be found in the k2 item.