The Internet Archive is the largest web archive in existence.
The Internet Archive is open for all users.
It is run as a private initiative driven by volunteers and sponsorships, and has been collecting web materials since 1996.
The Internet Archive has a policy of respecting if website owners mark their websites with "do not archive this website" (usually using robots.txt; a small file specifying access rules for automated web crawling). Therefore such websites will not be accessible in The Internet Archive (as opposed to Netarkivet which follows a legal deposit act where all published content must be stored for posterity at The Royal Danish library).
Moreover, The Internet Archive respects such wishes from website owners retroactively. If a website owner decides to add a "do not archive" clause to their website, it stops being accessible at The Internet Archive.
Web pages that do not have a "do not archive" clause (and are not technically impossible to archive) can be archived directly by any user with The Internet Archive's "Save page now" service.
The Internet Archive: https://archive.org/