It depends what your needs are, you haven't given any details. I'm not aware of anything with the JSONBin.io API (frankly I don't like their API, I think it should be more RESTful).

CouchDB is in many ways very similar to JSONBin and self hostable. I have also written small services that use SQLite's JSON support and wrap that up in a fairly restful API (e.g. POST a document to a list adds an item, etc.).

Another option is to use something like PostgREST, which I get isn't at all what you asked, but in this day and age you can just ask ChatGPT to make you a schema and it takes 2 minutes to prototype a REST API.

Really it depends if you have preferences for what datastore you use, as that is the more difficult bit when self hosting.