Do you mean how are people making archive links? They go to archive.is and provide a paywalled link and the website archives and displays the content. I can't tell you how they get around paywalls or how archive.is has managed to not get shutdown, but that's how it's done.
Add archive.is in front of it
https://archive.is/https://www.newyorker.com/news/annals-of-...
If you get an nginx page (I seem to get one pretty often), you can try archive.today, archive.li, or any of the alternates in the URL section on https://en.wikipedia.org/wiki/Archive.today
If the article has already been archived, you can select one of the snapshots which the archive site will show you.
If it hasn't, click to archive it and wait ~5 minutes for it to finish. You'll get access to the snapshot and a URL you can share.
It appears to be a rate-limit mechanism of some sort specific to a fingerprint. Clearing cookies for archive.[is|vn|fo|md] may (or may not) get you past it.