It's an iffy one, Fluffy.
I spoke to a representative of Business.gov.au a while back, and the first reply was that no, it wasn't necessary. Something he said made me question him further; did he mean that businesses using eBay as their platform truly didn't need an ABN, or to display an ABN?
At that, he qualified what he'd said. He'd assumed (and apparently it was the general assumption at the time) that people who sold on eBay were primarily doing so like a garage sale - getting rid of their own items, putting things up for auction. After a bit of a discussion, he ended up by saying it should be the case that a PROFESSIONAL seller ought to display his/her ABN on their website/webpage, irrespective of the platform, but that it would probably need to be clarified in the Act.