Why would you want it to be so small? How do you do the printing? Why not just use a literal diary?
Typewriters have some psychological usefulness. It's single function-focused (writing.) And objects which have strong emotional value can be very desirable too - hop over to Instagram.
And this one .. can be easily slipped in the pocket, like Nintendo DS.
With e-ink, even better. Longer battery life too.
You might be able to hook it up to a larger monitor.
As for printing. Micro-printers aren't new, some are as slim as rods. Abhorrently expensive though. Or alternatively that Polaroid pre-inked paper. Or just wireless printing.
One of the best advantages of storing writing in a computer is preservation. And if there's a tree-like elegant notebook installed (CherryTree is quite nice) even better.
My preferred size would be 8 to 10 inches. I think 10 inches would be a sweet spot, for me personally. Especially since I would want to use the eink screen as a reader as well as a screen for editing. You'd lose pocketability, but gain some additional flexibility and greater readability.
My wish list:
Eink screen, 360 hinge so the thing could be used as a book, backlit keyboard as a stretch goal.
Super user friendly & distraction free writing interface, but with Linux underneath and maybe a checkbox in advanced options to open the shell / vi / emacs for use. One very nice feature would be allowing split screen between two editors, or the editor + pdf / epub. Removes the potential distraction of looking away from the editor when you need to reference something. But no web browser, since that would encourage procrastination, even on a slower eink screen.
Wifi with automatic backups to the major cloud providers or user specified private servers. Plugged into USB and it shows up as simple storage.
Anyways, thanks for the fun idea which led to some fun thoughts. I hope we see something like this one day.
https://en.wikipedia.org/wiki/PalmPilot
Note btw that it ran on 2xAAA...
I say this because i've found recently i can focus and produce much better without a keyboard, but rather a pen and paper. George R R Martin uses DOS/word perfect, IIRC. Many have used typewriters, pens, pencils, etc etc. I find a writer's tools to be interesting in themselves.