There are no inoculation requirements for visitors to Australia, unless you have come recently from a list of rather unlikely countries. If you have visited somewhere off the beaten track on your way, check with an Australian embassy prior to travelling. Of course, medical attention, if required, will be expensive in Australia, so some form of health insurance is required. The main certificate in such cases will be a yellow fever certificate.
Some countries have reciprocal arrangements with Australia for health cover. If you check this situation when you apply for your ETA, you may find that you are already covered for your time in Australia.
Hospitals are excellent and the ambulance service to get you there efficient all across the country. In remote areas helicopters and aircraft will be used. The risk is snakebites, crocodiles, insects and sharks. However despite various very high profile deaths they are very rare and most incidents such as a snake bite if dealt with properly are not life threatening.
Be careful in the bush and do not travel alone in case of a snake bite and teak mob lie if you have one. Watch the Australian sun and avoid skin cancer, remember the slogan “slip, slap, slop” i.e. put on a shirt, slap on a hat and put on a high factor sunscreen.
There is a risk of Dengue fever and caused by mosquitoes that bite in the day time. This really means that you need to consider using mosquito repellant 24 hours in areas where Dengue fever exists.