While you can't save to localStorage as my sibling commenters have shown,
greyface- down below in the thread posted a version that saves to the hash fragment of the URI. Saving to the (Data) URI has a benefit over localStorage of allowing you to save by bookmarking, which also enables you to save many notes, not just one.
I code-golfed greyface-'s code and made the text cursor autofocus on page load:
I code-golfed greyface-'s code and made the text cursor autofocus on page load: