a public good is air.
it's non excludable, because basically everywhere you go, unless you chain yourself to the botton of the ocean of bury yourself underground, there will be air, and there is not forbidding you from breathing it.
air should not be provided by the gorvernment, because even if we were technologically advanced enough to control air and atmospheric pressure, the gornment would be giving out rations of air to us? that would be preposterous and chaotic. if it became a private industry, it would literally mean "if you have money, you don't die" which is inhumane.