I have attempted to store all my tweets in a WordPress blog so I can browse it as well as create my own searchable backup copy.
I just came across another cool service that lets me download my tweets in one shot as a PDF ebook.
You just need to sign in with your twitter account at TweetBook.in. Then you have the choice to download all your tweets or just your favorites as a PDF.
If you have a lot of tweets and if it takes longer than 10 minutes for the PDF to appear, make sure you logout and log back in.