How can I find the actual file referred to in a getmedia URL?

You can query the database to get the reference you need.

SELECT '/your-site-top-level-media-folder/media/', L.LibraryFolder, F.FilePath, F.FileExtension--, L.*,F.* 
FROM [Media_File] F 
INNER JOIN [Media_Library] L 
    ON F.FileLibraryID = l.LibraryID
WHERE FileGUID = 'GUID'

I can find the row in the Media_File table, but the file name it shows does not exist anywhere in the CMS directory tree. And I'm not even sure that's where I ought to be looking. How would I find out what this site's top level media folder is? Is that a setting I can look up, or is it maybe in the web.config?

It's in the settings of the Kentico instance under Content -> Media -> Media Libraries Folder. The most common configuration is the ~/<site name> folder.

That folder is also ignored in some source control setups, since it has many large assets, so it may not be in a repo.

If you still can't find it the only other place would be to see if they have an external storage provider like I talk about here on my blog.

It turned out the site media are stored in an AWS S3 bucket, which was not an easy thing to learn from the CMS settings and configuration.