Don't worry, everything will be back up and operational soon
Folks from all around are reporting issues downloading apps and updates from the Play Store today, receiving an error "403" in the process. It's safe to say at this point that Google Play is both herping and derping, and it should be sorted out soon. Here's a quick explanation of what that 403 error means, technically:
Generally a 403 error means your request was "forbidden." That doesn't mean you did anything wrong, just that the server (in this case Google's Play store servers) were programmed to deny the request you sent. Typically, you see a 403 error when you try to browse the file structure of a server that wasn't configured to allow this sort of access.
What's likely happening with Google Play is because of traffic shaping, where the normal request was altered and the server gets a request that it cannot allow. There's nothing you can do to fix them, though some people have had luck deleting their Google account and adding it again. The best thing you can do is wait it out, even though that's something nobody wants to do.
We go through these sorts of widespread errors from time to time, and they always get sorted out quickly. Hang in there.