I need a key-value storage in a simplest form we can think of. Keys should be some fixed-length strings, values should be some texts. This key-value storage should have an HTTP-backed API.

That's basically it. As you can see, there is no big difference between such storage and some web application with some upload functionality.

The thing is - it'll take few hours (including tests and coffee drinking) to write something like this. "Something like this" will be fully under my control and can be tuned on demand.

Should I, in this specific case, not try to reinvent bicycles? Is it better to use some of existing NoSQL solutions. If yes, which one exactly?

If, say, I'd needed something SQL-like, I won't ask and won't try to write something by myself. But with NoSQL I just don't know what is adequate and what is not.