Is this possible? I've heard that there's an extension or something for Vista's Media Center, but I can't figure out how to enable this in Windows 7.
It hasn't been released for Windows 7 yet. I imagine any attempts to get it working are completely unsupported.
Keep an eye on the Netflix Blog for announcements.
Try to see if MyNetflix works in Windows 7.
This is possible using PlayOn, which supports Windows 7. You don't even need PlayOn running on your Windows 7 machine though; it just has to be running on any machine on your network.
It registers itself as a UPnP Media Server, which any UPnP client (which Media Center is, I am almost positive) can access to play your Netflix, Hulu, YouTube, etc etc.