Numerous journals offer access to researchers, campuses, etc., through various systems. These are either individually authenticated, or are offered through campus-wide systems.

My understanding is that Sci-Hub works through "donated" access -- either credentials or inside-the-firewall systems -- through which it can request papers. There are other options for feeding papers into the system as well, and once they're inside, they're then available. Given that most disciplines tend to have a few highly-cited papers, and a standard Zipf / Power curve, starting with the most relevant papers and working down will get you a lot of coverage.

You've still got to keep up with new publications, but that's tractable.

Elsevier would have to deny access to entire academic institutions, or highly-placed researchers. Given they're already in a hot seat, that's a risky proposition.