I just finished attending the ETMOOC Twitter session and remembered that Twitter announced recently that you could request an archive of all your tweets.  Turns out this is quite easy to do.  On the web version of Twitter:

1. Go to your settings (under the cog icon in the top right)
2. Scroll down to the Your Twitter archive section.  
3. Click on the Request your archive button.

   A link will be emailed to you when the archive is ready.  I received mine very quickly as I have not tweeted much.

For an archive of other people’s tweets, the Library of Congress is the custodian.  They have just announced  that they have finished archiving 170 billion tweets and will now focus on making the archive accessible to researchers.