As for a story, if you type into the submission box you can save it as a draft.
And how long are these PMs that you are wanting to save as drafts? PM's are supposed to be short. They are also not the way to submit a story to the site.
While I do disagree with Zeb that Private Messages are meant to be short (which should surprise nobody who's ever encountered me), there is no way that I'm aware of to save it on the site in case of power outage or internet outage.
What I had taken to doing (before I disabled my PMs completely) was write it up in a document. Notepad that came with the computer if you have nothing else. Then, once you have it typed up, copy and paste in. Hit submit (and hope like Hell the internet connection doesn't drop again before it can upload [fuck Suddenlink]).
I don't know exactly what the limit is on size, but I have hit it and had to send two messages to get it all in. However, I doubt you'll reach it as that particular PM would have been around five or six screens on the story side.
***shrug*** And I find myself particularly unmoved by TLDR types. If I think it's worth saying, I'll struggle through nerve pain that makes it feel like some asshole replaced all my keys with heated needles hooked to electrical current. All they have to do is just not read it. Trying to bend me to their will by telling me to type less just makes me aware that they are not anyone I'm ever going to get along with anyway since no one else will ever be as important to them as they are to themselves.
One other thing, when you send a PM it does get put in your Sent folder. Of course that all happens on the sight where if you type it into notepad or some other text editor and save it.
And the last time I had an internet connection go down was well, just yesterday and what I was typing on the screen of my computer was still there. When the internet came back, a few seconds later, I just pressed Send again.
And if your on you phone, switch over to LTE instead of WIFI.