There are a couple existing in-product methods for getting exports:
Unfortunately there's no API-friendly method for either of these currently. I believe the reasoning is that most sites won't often be doing export operations (so manual is ok for most).
We've got an open feature request for adding an API endpoint here. For more information about how Atlassian prioritizes feature requests made on jira.atlassian.com, check out this Community post.
There's also experimental scripts that can help with a full site backup: https://bitbucket.org/atlassianlabs/automatic-cloud-backup/src/master/ Based on the backups generated by these scripts, you could then extract individual items with some additional effort.
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