First of all I don't think Obama is suggesting setting up a health care monopoly. I think he's suggesting supplementing the private market with a government-provided option.

As for how health care is different from everything else, it's very expensive sometimes, and society has kind of decided that everyone deserves a basic level of care regardless of the ability to pay. By that I mean people will generally respond negatively to a statement such as "If you can't afford it, you can't expect to receive treatment when you break your arm." In contrast I think it's pretty reasonable to say "If you can't afford it, you can't have a car."

I don't think the government should run a health care monopoly, but there are significant flaws in the market system that require government intervention in many cases, and health care is certainly one of them.

People will respond negatively to "if you can't afford it, you can't eat". But the government doesn't run farms and groceries, not even by "supplementing the private market". They give you food stamps. Why can't the government give you healthcare vouchers? Why do they have to set up their own insurance plan?

Like I said health care is sometimes very expensive. Way more expensive than food. That's why people are willing to pay for health insurance, and why insurance companies don't even offer "starvation" insurance that would, say, let you get all your groceries for free with some small co-pay. There's not enough variation in food prices over a range that we care about for our risk aversion to kick in.

So you are comparing food to insurance. Maybe part of the problem here is that people keep conflating government insurance with government-run health care. They are not the same. What's happening here is that the government is seeing a product that the private market is not providing, but society has deemed desirable for everyone to have. That's a pretty clear prescription for a government option (like USPS).

Why does the government even have to run an insurance company though? Why can't they give you insurance vouchers?

"Starvation insurance" is indeed a bad idea. So is "insurance" for routine care. It costs a relatively predictable amount of money to have a routine annual blood test and physical. Why can't I just pay that money myself, instead of paying an insurance company to pay that money for me? And if I can't afford it, instead of having the government set up an "insurance" company to buy me doctor visits and a dental plan, why can't they just give me vouchers for that too? Save the "insurance" for stuff that actually makes sense to insure against--things like hospital visits and so forth.

Why will a government-run health insurance plan perform better than existing non-profit health insurance? It seems pretty inevitable that tax dollars will end up being used to tip the scale in its favor, leading towards a single-payer system, which I think would be a bad thing. Monopsonies tend not to lead to the best outcomes.

I believe the Netherlands does something like this.

I'm not sure how accurate it is, but the description I saw of the Dutch system is that everyone has exactly the same plan for exactly the same costs, but the gov't does it through the fig-leaf of "private" insurance companies.