The world has ended due 200 years ago to a nuclear war between the US and China. The resultant radioactive fallout has made the surface very difficult to inhabit, and civilizations struggle to stay afloat in the wasteland. "Vaults" (huge pre-war underground shelters) hold the remnants of the pre-war population in them. You are one of these inhabitants.
On your 18th birthday, you wake up to find out your only surviving family (your father) - the vault's resident doctor and researcher - has left the vault and escaped outside to the wasteland. He has left a letter apologizing to you, and telling you he needed to leave for a crucial "mission".
Vault dwellers are not allowed to leave. If they leave at any point, they are never allowed to return. If you do not leave now, you will never be able to in the future.