Update: The service seems to working properly now.
The good news: Microsoft's new cross-platform Game Room service is now available. It's free to download and gives you access to 30 different classic arcade games that can be played on Xbox 360 or PC. The bad news: You can't actually play any of those games. Upon the initial Game Room load, users are prompted to download at least one of two game packs, after which the service should load up the games and let you start playing.
Unfortunately, Game Room simply hangs on the loading screen, allowing you to enjoy the retro music playing in the background, but not the actual retro games. Major Nelson has stated via Twitter that Microsoft is aware of the issue and it is "being investigated." He reminds users that the Game Room application itself is still available, and it should be okay to download it now and wait for the issue to be resolved. We'll update this post once we hear any new information.
Update: As some of our commenters have pointed out, there is a workaround. First, sign in with an offline profile or disconnect your internet source (ethernet or wireless). Then load Game Room. It should get past the initial loading screen. Once everything is loaded, you can then sign into Xbox Live and all the games should be accessible.