If you are a sole trader, have a state registered business name or an Australian registered company, you are eligible to register a .au domain name. You can read auDA’s policy on domain name eligibility and allocation at [auda.org.au...]
I got this for you from one of the registrars sites.
"To register a .com.au or .net.au domain, you must be an Australian registered company, or business with a registered business number (ABN, BRN, BN). You will need to supply this number during the order process. The domain name you order will need to be an exact match, acronym or abbreviation of the supplied business registration, or it should bear a close and substantial connection to the domain name. Domains can be two to 63 letters in length. Words can be separated by hyphens but not spaces."