Author without institutional or professional affiliation
This is an unusual situation, but when it arises giving your personal address as the "affiliation" is an acceptable approach. (Whether you will get a different reaction from journals and reviewers after doing so is another question...). Note that many journals still refer to this field as the "address" rather than "affiliation", which answers the question for you ;-)

I've seen personal addresses used in papers several times in various fields; most frequently in avian ecology, a field with a strong tradition of independent local researchers. However, it crops up in various places to a smaller degree.

In some cases, I know it's a retired academic continuing to publish (without an "emeritus" status) while in others it's almost certainly "spare time"-type research. I can't speak to how common it is outside the sciences, but I've seen a number of history papers with what are almost certainly personal addresses on them, so it's certainly a known practice there.