The ADF doesn't have a conventional prison per se, but what is known as the Defence Force Correctional Establishment (DFCE).
The DFCE is based in Holsworthy and run by service police from the three services. The purpose of DFCE is not punishment or to lock up criminals, but to rehabilitate and retrain errant service members whose services can still be retained. It is a very stressful and unpleasant environment, but detainees still receive lessons, conduct PT and can receive recreation periods.
As such, the average detention is only about two weeks, and detainees still receive pay (albeit recruit-level wages). If a service member messes up heavily enough and his services are no longer desired, he won't even go to DFCE, he'll just get kicked out of the ADF. Members of the ADF who break civilian law (as opposed to military law) are liable to end up in civilian prisons.