Medicare is the Australian public healthcare system, what did Australia have before it existed.
Prior to Medicare was Medibank, initiated by the Whitlam Government and one of the policies that resulted in the double dissolution. It was eventually implemented in 1974.
Prior to that, there didn't appear to be any standardised public funding (except for QLD which apparently did have a universal free health care program).
The following Wikipedia article should give you a good place to start for further research.