Education doesn't, but police and fire definitely do: police because beat cops reduce crime for everyone, so they're a natural monopoly. Fire as well, since stopping houses burning down helps stop the entire neighborhood (especially when you have close-together row houses) burning down.

Healthcare is arguably a public good, at least as far as vaccinations and contagious diseases go, since herd immunity is a large way to fight them (especially for contagious diseases for which we have 80%-90% effective vaccines). Not so sure about other illnesses (like cancer) though. I'm personally for socialized medicine for other reasons (humanitarian), but I agree that the natural monopoly argument is not applicable there.