As discussed in: Graduation, site closure, and a clearer outlook on the health of SE sites, just about the only criteria remaining to keep a site open is having enough users (three minimum) willing to step up to Moderate♦ it.
If a site is in decline and doesn't have enough users willing to take care of it, we will typically issue several calls (through a featured meta post) to try and recruit more interest. Failing that, we will post a final site-closure notice announcing the closure date. This will appear as a 'system message' plastered across the top of every page on the site — all the time referring users back to a meta post where they can express interest in keeping it going.
I can't imagine a site going from functional to "final notice" in less than a month; more typically the effort would persist three to five months… or more. Typically you can tell when a site is no longer working because view counts drop, questions go unanswered, and meta community issues go unanswered or never addressed. There is no particular system to reach out to non-users or disconnected users who are no longer participating in the site. If a site is closed, its content is archived through a "data dump" (search meta), or you can typically find specific posts through the Wayback Machine if you are wondering why a link to a post only brings up the final closure notice.