You could use something like archivebox as that saves the whole page, or you could use Waybackmachine and force it to save the page via an add-on.

You could also setup your own yacy index and everytime you find an interesting site you could add it to yacy.
But this is kind of not what you are asking for. Archivebox is probably the closest, or using squidcache and literally caching every url you go to. 😅