UPDATED: As far as I know everyone on Twitter can now download an archive of all their twitter activity, there are a few neat things you can do with the data once you’ve got it. For instance, you could upload it all to your website (or your Google Drive) make it public and searchable. I had this setup for a while, but removed it a few years ago.
All my tweets dating back to when I first signed up in June 2007 were available for your delectation. A mere 32,000 tweets at the time of writing. By using the unconverted root file “index.html” from your archive and the attendant CSS files etc you effectively get a browsable website of your Twitter archive. It looked something like this.