Goodness!

The essential, first-order or pure concept of "public goods" is whatever we benefit from that is "non-excludable" and "non-rivalrous." That means you can use it without paying for it, and if you use it, it's still there for others. An example, at least for the time being, is the oxygen in the air. Back in the good old paleolithic, there was a lot more of that. Basically, the land and the water and the plants and animals were there for the taking, and there was usually plenty so rivalry was uncommon. Of course, this only worked within your own tribe -- sometimes people of different tribes tried exclusion and rivalry, and even killed each other. But you get the idea.

For better or for worse, nature for the most part doesn't work that way any more. Land is now a commodity - it is owned, bought and sold, and the law says you can exclude people from your property. The products of the land are owned as well. Most public goods nowadays are actually created by humans, and they may also be thought of as goods and services with very large positive externalities. A classic example is a lighthouse. It was worth it to the local port authorities to build it and maintain it -- perhaps a joint investment of merchants, and/or navigators, or the king who stood to gain from taxing them and dispatching his navies more safely -- but every sailor benefits, even if they're just passing by. A comparable example today is the Global Positioning System. 

 

The point is, the Free Market™ could not create a lighthouse, or the GPS, because there's no way to make people pay for it. Most examples are not quite so pure. The market would produce some amount of the good in question, but less than the optimal amount for society as a whole. That is because the good or service has large positive externalities -- benefits to society beyond their market value. A classic example here is education. Affluent people would send their children to private school, if there were no public education, as they have indeed done since Aristotle tutored Alexander. 

 

However, there are major benefits to society if every child is educated, because nowadays the economy needs a workforce that is literate and numerate, and a lot of people with more specialized skills, be it carpentry or nuclear physics. The road and highway network is another example. People would build some bridges and perhaps roads on which they could charge tolls, but the transportation network that the modern economy requires is far too extensive and interconnected for the market to produce. There is no nation on earth that doesn't have a government funded road network and educational system, and the richest countries have the best ones. But guess what? In order to provide these public goods, the government needs to make you pay taxes.

Medicine also has huge positive externalities. (Of course it has negative externalities as well, including carbon emissions and notably, a huge quantity of plastic waste.) A straightforward positive externality is infectious disease control. Preventing or curing infectious diseases prevents them from being transmitted to others. This is an immense benefit to society that goes far beyond the direct value to people who are vaccinated or treated.

Another positive externality is that people with curable or ameliorable sickness or disability who might otherwise not be able to work can remain in or return to the labor force, and so improve the economic well-being of themselves and their families, and the productivity of the entire economy. They can also better take care of their children or other dependents, maintain their households, volunteer in the community, pay taxes, give to charity, and whatever other good things healthier people are better able to do. And obviously it is distressing to people when their friends and loved ones are sick, disabled or in pain. It may even require them to give up other productive work to care for someone else.

Even if they aren’t directly affected personally, many people are disturbed by the thought that people who are in dire need may go without care that could cure or succor them. It would constitute a great public offense if people were dying at the doors of the hospital because they could not pay, as would the sight of seriously ill or injured people on the streets. Or rather, we do see such sights all the time, but homeless people’s illnesses and injuries are mostly psychological, and for some reason the public seems willing to tolerate that. We’ll get to this problem later.

For now, however, one argument in favor of universal health care (not the only one) is that it will more than pay for itself through these positive externalities. And you know what? That is actually true.