Singleton object, so I can use object for site configuration?

I would like to create an Object for other Objects in the same site to use for configuration. In addition to being more accessible from microservices, using an Object for configuration would also make it easier to build a UI and handle permissions for editing it.

However, I’m not seeing any way to create an object that must have a unique site ID, or a singleton object, or even a “site” system object that I could add fields to.

Is there a way to use Objects for site-specific configuration?