The reason I want something like this is so that I can easily set something up on my personal server then I could go to my bank's website, view my previous statement, and easily say "Send this to my personal file locker on my server for long-term storage", regardless of what computer or mobile device I'm at. It would also be possible to do the same thing with utility bill statements, tax papers, etc.
The primary reason I would want something like this is because third-party services come and go, and I don't want my data disappearing with them. Many utilities and banks want to go paperless, yet I want a copy of the bill digitally signed by them on my server where they can't retroactively change it without my knowledge. There have also been numerous legal loopholes where if the service is on a server not owned by that person then that person has no say if the authorities feel like accessing their data. It would also be great as a NAS-like backup and sync between personal computers, without any of that data ever going outside of your property.