Typically included in your benefits package as part of the job. You don't have to give up Australian citizenship to get an E3, so if necessary you can fly back to Australia for treatment.
If a sickness/injury is bad enough to need to fly back to Australia, there's a pretty good chance you won't be allowed on a plane.
That's fine. You're ok with taking that chance, I'm not.
As far as the ATO is concerned, unless you discontinue both, you're considered an Australian resident for tax purposes.
Even though the US considers you a resident alien, the ATO requires you to at least look like you won't be coming back for >1 year.
After ~3 years the IRS considers me a resident for tax purposes. The ATO only cares about my income because they want to collect HECS payments.