Could the European Union accept a member state or states that are not geographically part of Europe (e.g. Australia) under the current rules?
Only European states are allowed to join the European Union. Article 49 of the Treaty on European Union (as amended) provides that:
Any European State which respects the values referred to in Article 2 and is committed to promoting them may apply to become a member of the Union.
Of course there is a certain amount of ambiguity of what it means to be European and it can be defined in cultural, historical and geographic terms. Cyprus is a EU member state even though it's geographically a Middle Eastern country, and Turkey is a candidate member state even though only a small part of it is in Europe, geographically speaking. Both countries are also members of the Council of Europe, the membership of which is also limited to "European States".
In contrast when Morocco applied for EU membership for 1987 it was rejected on the basis that is was not a European country. In that case there was no ambiguity.