You can die in:
-> Missions
-> Traveling Places
-> In Game Places (E.G: Villages & Traveling places)
-> RPG Places
Areas you can't die in:
-> Kage's Offices (Not during War)
-> Streets & Housing (Not during War)
-> D-Ranks
-> Out of Game places (E.G: Shinobi City, any place that isn't a Village, except of Death Arena)
-> Academies & Dojos (Not During war)
Akatsuki
Akatsuki is a special organization that has special rights. They are able to kill in areas that normal shinobi can't and can die as well. They are limited to the number of deaths they can have before being kicked out though. Look here for details: Akatsuki Rules
Missions
As explained earlier, missions are a key source of obtaining money. However, by doing missions, you can also gain life and chakra too. 500 Life and 500 Chakra max though/ it increaes when you move on in rank. Missions are posted by your kage and can be requested by PM-ing him/her. The rules to and types of missions are posted here.