Suppose you are insuring yourself against some event type E with an insurance company with claims ratio, say, 0.75. This means that the company pays out 75% of the net premiums in claims. On its face, this seems even more irrational than gambling at a casino—as far as I can determine with a bit of internet "research" (see for instance here), a casino tends to pay out a larger percentage of what is paid in than 75%. It seems irrational because unless you have special information about your case (in which case there are some integrity questions that might be raised), you can expect to get back 60% of what you put in.
But there is a crucial difference. One typically insures oneself against adverse circumstances. In adverse circumstances, money may well have higher utility than it does in normal circumstances. For instance, if your car is stolen and your employment depends on having a car, the value of having an amount of money sufficient to purchase a car is significantly greater than the value of having that amount of money in normal circumstances where you already have a car.
This suggests a rough heuristic: it is rational to insure yourself against E with a company whose claims ratio is r for a claim amount c only if the utility to you of receiving c in case of E is equal to the utility of receiving c/r in case of non-E. (For a better estimate, one would have to take into account potential investment returns on the money that would have gone out in premiums.) For an egregious example, extended warranties (a species of insurance) have a 0.43 claim ratio (UK data). Thus it makes sense to get an extended warranty for a $400 TV only if getting $400 in the event of the TV's breaking down has as much utility to you as getting $400/0.43=$930 under ordinary circumstances, which is unlikely to be the case. (Though it might be if you expected to be low on cash and your well-being is strongly enough tied to having a TV of the relevant price-level.) But in the case of, say, car theft coverage it might be worth it if you would be unlikely to be able to pay for a new car of sufficient quality and your well-being strongly depends on having a car of that quality.
Interestingly, I think it follows that it shouldn't be worthwhile insuring luxury items, unless (a) you wouldn't be able to afford replacing them otherwise and (b) your well-being is tied to them to a high degree. But it is probably vicious to have your well-being be so tied to luxury items.
OK, except for the thing about luxury and vice, this is stuff that's no doubt obvious to every economics student, but it wasn't obvious to me, and the heuristic is kind of handy.