Russia could not properly invade Australia right now. While Russia does have a large, competent land-force that has a lot of firepower, and although it is rapidly modernizing its air-force, it has almost no Navy. The Russian Navy has almost no air-craft carriers (no modern ones at least), and few ships equipped to transport and supply an army across an ocean. The Russian navy is geared more towards Defence, utilizing surface-to-surface beyond-horizon anti-ship missiles.
Although Australia's Navy is incredibly weak (6 of our 12 most modern destroyers are in no shape to actually fight), we do have a competent airforce which could mount a coastal defence (although, admittedly, not for long given Russian superiority in air-power and sheer numbers). An invasion into Australia would not go well for the Australians - we simply do not have enough tanks to hold off an invasion if they managed to get a lot of troops to our shores.
Thankfully, the US would almost certainly step in, and Russia knows this. With the US fleet on our side, Russia would have no chance of actually crossing.
If Russia started to build up its fleet, it might pose a threat to the US Navy. But as it stands right now, the Russian Navy is under-funded and the Russian government currently has no plans to build a truck-load of new air-craft carriers. Russia is building its army up primarily for defence and local power projection. It has no plans to invade the rest of the world, it has no ambitions beyond kicking Eastern Europe around, so why would they waste money building up the capability to launch force overseas? Unlike the Americans, the Russians have absolutely no desire to play "Globocop" and no desire to "conquer" Australia.
So if Russia tried to invade Australia, we could hold out by ourselves for a while, but eventually we'd be overwhelmed. If we still have the US on our side, the Russians couldn't actually get here. Period. Their Blue-Sea capability is nearly non-existent right now.
Same goes for China: China's got a big land force, a pretty scary air-force, but it's Navy is extremely weak. Unlike Russia, however, China actually DOES plan to bolster its Navy, and build many, many new Aircraft carriers. If anyone could/would invade Australia, it would be the Chinese, but I somehow think that's highly unlikely. The Chinese generally have no desire to "conquer" other lands. They're happy with the land they've got, they've got no desire to attack and consume other nations. If China really wanted to, it could have claimed Vietnam, Burma, Thailand and Laos - but it hasn't. And it's not going to. The Chinese have a very clear-cut concept of "what China is", and they really don't care for taking over lands that they consider "not china".
EDIT: Okay, assuming that they actually magically DID manage to get here (somehow), we'd be in trouble. Australia has about 50,000 - 70,000 "trained" troops, and about 20,000 of them are just reservists. To be fair, our troops are pretty good shots and they know what they are doing. Unfortunately, we don't have the money, the oil, the supplies or the manpower to last very long. We've got about 50 or so M1 Abrams tanks, given to us by the Americans, but they are the "export" version and they aren't actually very good. The Russians, if they could get here, could deploy thousands of tanks. Doesn't matter if they aren't as good as the M1 (although some of them, especially the T-90 are pretty damn scary), hundreds if not thousands of tanks CANNOT be stopped by just 50 or so tanks. Australia is also a rather sparse country with few major centres of population and manufacturing. Once you capture the 5 cities that matter (Sydney, Melbourne, Brisbane, Perth and Adelaide), the Russians are laughing - all they have to do is blockade six cities, a couple of highways, and they've essentially cut our population centres apart and frozen our capacity to fight back. Russian Tanks could just drive around the outback at will - Australia's still mostly desert and shrub-land.
So, to sum up:
1) We simply don't have the numbers to fight off an invasion
2) We don't have the equipment
3) We don't have the money
4) We don't have the amount of cities and manufacturing centres necessary
5) Our population is too concentrated and our transport is contingent on a few major highways
6) Our Navy is incredibly ill-equipped to stave off any invasion attempt
But none of that matters since the Russians will never get here. They don't want to get here, their navy is also weak, and the US is allied with us. So there's nothing to fear.
But yeah, if they did invade, we're screwed. We simply do not have the means by which to hold them off. I'm sure we'd give a very spirited defence, but we'd be doomed without international help from either the UK or the US.