On a regular basis, my employer offers workshops and sign-up drive for long-term care insurance. I have never bothered with it, and I do not think any of my colleagues has. Yet, it makes perfect sense to participate: the likelihood that one needs long-term care, either with a visiting nurse or in a nursing home, is high and expensive. Yet, this is an eventuality that is so far away that it is discounted heavily from our minds.
Pierre Pestieau and Grégory Ponthière say the problem goes beyond this discounting. It is a multiple equilibrium situation. Currently, very few people buy this type of insurance, which makes it more expensive, hence few people buy it. If there were a larger market, it could be sustained by lower costs.
This seems to be a simpler explanation than what people have come up for the lack of an annuity market or one for reverse mortgages.