If you download your Twitter archive it arrives wrapped as a static HTML page, which is not very useful for doing anything with, and worse: it requires the original account to be still active to do useful things like enlarge the images since they use t.co links.
So here's a Python script to convert a Twitter archive to markdown or other formats: https://github.com/timhutton/twitter-archive-parser
Now you can archive your tweets in any way you want.
@Feral3D @timhutton Using the above Python code it certainly sounds like it might be possible. More research needed though to be certain. If it is possible then it wouldn't really be a "history" as you'd be dumping all your posts in a short period of time. You might even hit snags with bandwidth on some servers.