How to archive your Tweets with the Wayback Machine
You can request an archive of all your Tweets from Twitter and upload that archive to the Wayback Machine for your Tweets to be archived.
To request an archive of your Tweets from Twitter see these instructions....
https://help.archive.org/help/how-to-archive-your-tweets-with-the-wayback-machine/
IF you want to save your tweets publicly, the Internet Archive makes this possible.
(And if you don't, email the archive at [email protected] with your request, see: https://help.archive.org/help/how-do-i-request-to-remove-something-from-archive-org/)