There are at least 1,500 species of bee native to Australia. Some are solitary (i.e., don't live in communal hives) and a most species sting. The interesting ones if you want to keep your own bees in your back yard are the stingless social kind that make honey. The most common variety found in Southeast Queensland are trigona carbonaria. These bees look more like ants than like a European bee. In the wild, they live in holes in trees, and their habitat is steadily reducing as trees are cut down. They are however very adaptable, and have been found in dark spaces inside houses.
It's easy to keep these bees in your back yard, but don't expect anywhere near the honey yield that you'd get from European honey bees. A hive produces at most 1kg a year. The honey is much runnier than commercial honey (the bees obviously do not reduce the water content as much as their bigger relatives do), and is hard to buy commercially, and expensive. It's real bush tucker, and known as sugarbag among Aborigines.
Trigona carbonaria are effective pollinators, and are especially good on indigenous plants, such as macadamias. Different plants require different modes of pollination, so don't expect the same insect to be good for everything. Biodiversity as with so many other aspects of the environment is important. These bees will not frighten away other pollinators, since they are small and not aggressive.
The bees are much smaller than European honey bees, so they need a smaller hive. They store their honey in different structures than the well-known honey comb. The inside of the nest is laid down in a spiral shape, and the honey is stored in little spherical pots made of a mix of wax and resins the bees collect from plants. If the nest is big enough, they store the honey far from the brood area, where eggs hatch and develop into new bees. This is a good strategy to protect the brood area from predators, who would mainly be attracted to the honey. A good hive design has multiple levels, so the bees can isolate the honey far from the brood area. This makes it possible to remove the honey without disturbing most of the nest.
They also are quite different from honey bees in details of their behaviour. I haven't observed mine doing the well-studied bee dance so they presumably have a different approach to guiding others to food sources. Although they form swarms occasionally, this has nothing to do with starting a new hive. My view is that these swarms arise from rapid growth in the number of bees under good conditions, resulting in the hive sending bees out too fast. When they return, traffic backs up, like an airport in bad weather. After this happens a few times, they appear to learn to moderate the rate of sending foragers out so the backup doesn't happen any more. At its height, a swarm can extend several metres into the air. Rather than stinging (and subsequently dying) to defend themselves, they rely on weight of numbers. If you threaten the hive, they swarm all over you. They deal with invaders by gluing them down with the mix of wax and resin they use as a construction material, then dismember them.
You can buy a hive and get it shipped to your home. I bought mine from Russell and Janine Zabel, who have a long history of rescuing bees from felled trees.
Once the hive is well established, it can defend itself against most predators and parasites. Provided you put it in a reasonable place with some protection from the worst of the sun (the nest material can melt if it's too hot), the bees are the ideal pet. They do their own thing, and you can ignore them unless you want to watch them. Harvesting the honey is a lot easier than for stinging bees (though I have seen reports that wearing a little protection is useful, as the bees can be uncomfortable to deal with if large numbers swarm over you, as they would in defence of their home). They don't need any special food: their foraging range is about 500m, so as long as there are flowers in that radius, they will look after themselves. This is an ancient species: similar bees have been found preserved in amber 80-million years old, so they've out-survived the dinosaurs.
A bee can be distinguished from a fly by counting wings. Flies have 2 wings, bees have 4. A wasp is very similar to a bee, except it's not vegetarian. It's useful to know these things because many of the 1,500+ varieties of Australian bee don't form hives, and can easily be mistaken for a different kind of insect, if you (for example) see one disappearing into a small hole in a tree that is obviously not big enough to be a hive.
There is no reason not to keep these bees even if you have a very small yard. The hive is small, and the bees will find their own food if your garden is inadequate. You would not, however, buy them as a paying proposition. A complete kit including a hive with bees costs around $300. The best reason to have them is to provide a home for a species whose habitat is threatened for no better reason than that too many people can't see the value in nature. If at some time in the future, humanity comes to its collective senses, these little bees will happily return to their natural habitat.