As a first approximation, the answer is simple: it can be done as long as there aren't too many externalities from house fires. If a fire in one house, unchecked, causes other homes to burn as well, mandatory fire services are the only solution. However, if the houses are far enough apart, as seems to have been the case here, there is no objection to making the fire department optional, except possibly the moral one that society should always come to the aid of people trapped in burning buildings, even if they are trapped there by their own laziness or selfishness.
A more interesting question - also useful for microeconomics 101 exams - is what to make of the offer to pay "whatever the cost" if the firefighters would come immediately. In the comments to the Volokh Conspiracy post about this story, Sasha Volokh gives - I think - the correct answer:
Of course, this is all well and good but how do you recover this fee from someone whose house just burned down?
The fire department could have said “$75 per unit of time for insurance OR
$10,000 if you need protection on the spot.” But the amount for spot protection
had better be pretty high, or else you’ll see people systematically forgo the
insurance and just pay on the spot. Just like with optional insurance generally,
you can see unraveling (or just inefficient development) of the insurance market
by adverse selection; only since the price of spot protection is going to have
to be lower than the actual cost of losing your house in a fire, the problem
will probably be worse.
UPDATE: Krugman heard about this story too, and he doesn't like it. He compares it, somewhat implausibly, to the health care debate. I don't see how that analogy can work: surely there is a difference between allowing someone's house to burn down and denying them medical care?