Aboriginal Australians did build structures, albeit temporary ones more akin to shelters than a building or hut. Think teepees but customised to locally available materials; mud, ferns, some rock, even whale bones as the framing. It's important to note, though, that this is to group Aboriginal Australians - we're talking multiple tribes in vastly different climates of a vast continent. Thus you have in the mix those Aborigines who built no structures at all; the nomads.

Are there those that are still standing? In a historical context no. These were always temporary structure. Those built by modern Aboriginal Australians still stand, but these wouldn't fall into the scope of your question.

Yes. Yes, they did. But, not many.

Since Europeans arrived in Australia, Aboriginal stone arrangements have been observed across the country: these include fish traps, monoliths, heaps and cairns, and circles.

One of these stone arrangements got itself into the news a few years ago when astronomers and historians started to theorise that it's an astronomical calendar. The Wurdi Youang stone structure (in modern-day Victoria, near Port Phillip Bay) was first noticed by Europeans when they first arrived in the area nearly 200 years ago. However, it wasn't charted until 1977, and was analysed only a few years ago:

Our detailed survey of Wurdi Youang supports the Morieson hypothesis that a series of outlier stones marks the position of the setting sun at the solstices and equinoxes. A statistical analysis shows that the likelihood of this occurring by chance is extremely low. Additionally, we find that the straight sides of the arrangement also indicate the solstices while the three prominent stones at the western apex of the arrangement, as viewed from the eastern apex, mark the point where the sun sets at equinox.

The indications are good that this is an astronomical calendar, similar in function to structures like Stonehenge in England, and Warren Field in Scotland, and Ales Stenar in Sweden, and many others around the world.

We currently don't know how old this structure is. Some people have speculated it could be older than the Stonehenge structure in England, but it may also have been created as recently as a few decades before European arrival.

A prominent example that hasn't been mentioned yet is Nawarla Gabarnmung in Arnhem Land. Gabarnmung is a rockshelter carved straight through a low stone ridge some 28,000+ years ago. The ridge was excavated until only a select view pillars were holding up the roof. A similarly constructed archway is thought to have been carved out over the path leading to the site, but it has since collapsed.

Here's a clip from the documentary First Footprints that will walk you through the site with a pair of archaeologists going over their findings with a Jawoyn elder.