In NSW you are entitled to lodge a claim against the "Nominal Defendant" (the Nominal Defendant is a scheme run by the State government for exactly this situation).

However, strict time limits apply. You need to report the accident to police within 28 days and lodge a Personal Injury Claim Form (available on the Motor Accident Authority of NSW website) within 6 months of the date of the accident. When lodging a claim upon the Nominal Defendant you are also required to conduct "due search and enquiry" in an attempt to establish the registration details of the unknown vehicle.