As
mithrandir pointed out, there are some rules regarding eligibility for domain names.
The
.au top level domain is managed by AUDA. This organisation don't sell domain names (they're not a registrar) but do enforce the rules.
There's a policy document (PDF) for the .au TLD
here. There is also a list of accredited registrars at the auDA
website.
The easiest way to comply with the business requirements is to get an
ABN (single trader). It one of the few things you can get free from the government.
