I'm trying to import my figma file as a data source. I'm getting an error that the file has not been published. It's not a file that I wish to publish publicly. What can be done?

Publishing your Figma file does not mean that the file will become public.

You just need to click on the Figma icon, select "Libraries" and publish your design file in your internal library, which is accessible only to you (or your team). The whole process is described in Add design system content.