TL;DR: You can back up pages to the Internet Archive by submitting them here:
https://web.archive.org/save. Save the fanfic!
It's been an ongoing project of mine to back up the fic archives of my fandoms the Internet Archive. Someone else seems to have done a thorough job with the Circuit Archive already, but I've done some work on The Hatstand, and there is more to do. Tonight I was working on a Man from U.N.C.L.E. archive that links to a lot of external works, some of which were hosted on personal sites that no longer exist and that were never backed up to IA. In the end, I had to shelve that task for another day, because it was too disheartening. I know that for as long as there have been fanfics, fanfics have been fading away, but it always makes me sad.
All this reminded me that not everyone knows that it is possible to submit pages to the Internet Archive. You can do it using this form:
https://web.archive.org/save. There are also browser extensions that allow you to submit the page you are viewing. I'd encourage everyone who hosts their fan works on their own site to back them up to IA. You never know what may happen. I'm not sure of the propriety of backing up others' work from their personal sites, but that is also possible. And anyone is certainly welcome to help out with The Hatstand.
(Note that once submitted, pages often take a while to become available on IA, which means that you usually need to come back after some time has elapsed if you want to check your submissions. I'm not really sure how much time is necessary, but a week seems to be enough. If you register at the Internet Archive, you can use a feature that backs up pages linked from the page you submit. I've found that this technique is not wholly reliable, though, so it's necessary to do a later pass by hand if you don't want anything to be left out. Also note that backing up pages from LiveJournal doesn't always work because of age protections and privacy settings.)