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:
> Webstorage is tied to an origin. 'data:' URLs have unique origins in Blink (that is, they match no other origins, not even themselves). even if we decided that 'data:' URLs should be able to access localStorage, the data wouldn't be available next time you visited the URL, as the origins wouldn't match.