The public system in Melbourne is very good. I have full private cover but chose to go through the public system when I had my son and would do it again in a heartbeat.
How pregnant are you?
I'm not sure how much it would cost to go to a public hospital with no medicare cover, but private would cost more. Add to that the private obstetrics costs (there are seperate chaerges fro private obstetricians in addition to your hospital stay)
Under the public system some of the options you can choose are to see an OB, OB& midwife, GP and midmife or team midwifery with only needing to see an OB on a few occasions.