Does the journal Physical Review require an institution in the affiliation field?
To my knowledge, there is nothing preventing someone from submitting a paper to a journal without a formal institution as part of the reader's address. You might have to write something in a web form such as "self-employed" or something similar if institution is a required field for submitting the form. 

However, it is fair to say that under most circumstances, if you are submitting a paper without institutional affiliation, it will probably be subjected to heightened scrutiny relative to a paper with more "traceable" affiliations. Unaffiliated authors are usually treated with some skepticism in the academic community, and papers by such authors tend to to be viewed as "crackpot" papers. (Granted, having an affiliation does not preclude the possibility of submitting such papers, but the likelihood is somewhat lower.)