That's an aspirational goal but there has been no practical methods proposed to actually achieve it.
I would bet that it could be done. It might not be cheap or easy, but I bet that it could be done.
What I am more dubious about is whether:
It can be done at the kind of cost that is considered acceptable by the public. I don't see budget numbers in the article.
It could be done without wiping out other species. One small mammal is much like another small mammal to, say, a poison (which the article is talking about them using today). If you want to wipe out rats without wiping out something else, you're gonna need a pretty darn selective method of killing.
Rats can be kept out once killed off in New Zealand. Rats are pretty much everywhere, and they got there because they're pretty good at hitching rides. I can believe that they might keep them off a small island with little shipping traffic, but the main islands receiving shipments from around the world?