The solution to this is... robots.txt. Otherwise your site might turn up in Google etc. Since it's archive.org that's doing the crawling and they respect robots.txt it won't get archived.
EDIT: Internet Archive started ignoring robots.txt in 2017: https://www.digitaltrends.com/computing/internet-archive-rob...
It says Internet Archive had already started ignoring robots.txt on US government websites.
Now (since 2017) they ignore it on all websites.