> right click on an item, click "copy public URL", paste it into the browser, and get an exact copy of that item

You have described Google Drive.

Not quite; it's not a direct link to the item.

Put <img src="foo.jpg"> into an html file, alongside foo.jpg. In the original Dropbox, if you opened a link to the html file, you'd see a webpage that successfully rendered foo.jpg. So you could use it as a static file host.