Across the Snowy Mountains in NSW there's dozens of cabins you can just go turn up and stay in, likely to have bushwalkers come by though if you stay a long time, they're public, and are from early 20th century when the place was agricultural then reclaimed to national park. Have to walk in with all food.
Also in the blue mountains of NSW there's a few caves, some set up in the 20s, in range of retreat centers, there's one yogi I heard of through the grape vine at a retreat center who was living in a cave and going on alms rounds to the local thai restraunt.
Probably wouldn't do this unless you've already had considerable experience with hiking / outdoors. And there's probably similar opportunities closer to your home & all around the world - where there's mountains. The blue mountains are basically the outskirts of Sydney though

And every buddhist tradition has some representation in sydney, often without the filter of california / east coast - it's the immigrant communities that bought it here.
Also I know that if you've sat a number of goenka retreats, depending on the center, some centers have accomodations where someone can sit the 30 or 45 day program, but you'd have to talk to a teacher there.
This is all probably a terrible idea and I'm just projecting on to you my fantasies ;)