Heller better describes the rights US citizens actually have.
Is that the 5-4 case? The constitution was upheld by one vote?
Yes. IMO, the only thing surprising about Heller was that it had 4 votes dissenting. Nevertheless, it's a Supreme Court decision, with this to say:
Held:
1. The Second Amendment protects an individual right to possess a
firearm
unconnected with service in a militia, and to use that arm for
traditionally lawful purposes, such as self-defense within the home.
[emphasis added]
While hunting is not explicitly mentioned, it's certainly a traditionally lawful purpose.