Only case I'm aware of where it's profitable as such comes about indirectly:
One such scenario is if, due to whatever legal situation, someone's going to lose their liquid assets but keep the house. In that case taking $10k out of the bank and putting it into a battery will at least retain some of that value, getting some of it back beats losing the lot.
Another is as above but with the reason being it gets them under the threshold to qualify for whatever payment (welfare). If buying a battery means they now qualify for a concession card or whatever well that could make it add up financially.
Another is risk mitigation for anyone running a business from home. Eg own two physically separate computers, have NBN with (say) Optus and use another network (eg Telstra) mobile as data backup, have backup power. No single failure of IT, communications or power stops the show, it's only things like burglary or the house burning down that remain a risk. In this case the battery could presumably be claimed as a tax deduction somehow. Tax saving + energy cost saving + whatever value is put on the risk mitigation = could plausibly add up.
Realistically though most installations I'm aware of fall into two categories:
Engineers, electricians, and other random people with an interest in technical things who just wanted one, had the money, and reasoned that once they deducted the energy cost savings of the battery over its lifetime, they were willing to spend the gap between that and the cost for the sake of having one. Bonus if they're a licensed electrician able to put their own labour to use.
The other is people for whom cost isn't an issue and who just liked the idea of backup power either for convenience or because they work from home, have an aquarium full of fish, or otherwise have something to lose if the power fails.