Last week I made available an archive of all App.net posts, which I had downloaded right before App.net shut down. There are now a couple tools to help process your own App.net posts from the archive.
Jordan Merrick has an iOS shortcut to download your posts:
Posts are made available as JSON dictionaries containing various pieces of information, such as the text and date created. All posts are downloaded and saved as individual JSON files to iCloud Drive at /Shortcuts/ADN Archive.
And Matt Bircher has a web app that can download your posts as a CSV file. If you have thousands of posts, note that both solutions are going to take a long time to run, because they have to download one post at a time from the archive.