If a government fails what rights and responsibilities do people have?

"Rights" are really just everyone passively agreeing that there are certain things you can do and certain things you can't. Without anyone to enforce them though there wouldn't really be any such thing. Without the government or its infrastructure (police, fire departments, road workers) people basically do whatever they feel like. Now while some people would probably loot and steal and riot (opportunists, people without much to lose), others would choose to try to bring structure back to society, and on the whole, people like structure so eventually a new government would likely emerge. To come back to your original question though, without the govenment no one technically has any responsibilities or rights beyond survival.