It was time for me to learn about websockets.
/u/Stuck_In_the_Matrix recently released a tool that provides a continous stream of comments. I read that stream and publish comments to my PubSub server, which this website connects to.
The underlying websocket server is being run as a service on a VPS I use for projects. I wouldn't suggest building anything you aren't willing to have break.
You can run it yourself, though. The code is available on github. You don't need to run the PubSub server to consume the stream from RedditAnalytics.com.