Well, there actually is way to get married without state involvement, at least in a few jursdictions: "
Common-law marriage" essentially means that if a couple has been living together and "acting married" (including representing themselves as such), for some given length of time, they can be recognized as having a "common-law marriage". Note that this is fairly scarce in the U.S., and the legal conditions are much stricter than the layman's idea of what the phrase means.