The platform supports server-side technologies the same way a C++ or Java application would support them: through HTTP (probably RESTful) interfaces that the server side of your app (if any) exposes. For example, your app might make a request to http://example.com/api/foo/bar/baz?param1=123¶m2=456, and might get back a JSON response that your app would then parse. It doesn't really matter which server-side technology you're using, because the API looks the same from the app's perspective.
If you're asking about PHP, Ruby, Python, Go, node.js, etc., running on the user's machine as a client-side app, no, that's not how Chrome packaged apps work (though in theory one could get one of those environments running as a NaCl module that then generates client-side code). Instead, you write code in a very similar way to how you'd write a pure AJAX app: in JavaScript, without navigation, with the content/data being fetched in a RESTful fashion from whatever server/service you use (or none in the case of something like a calculator or simple game that doesn't talk to the network at all). Incidentally, we're finding that very AJAXy apps are relatively simple to convert to packaged apps.
Check out all the samples at https://github.com/GoogleChrome/chrome-app-samples to get more an idea of the shape of the platform.
