Good question!
I'd give you some examples to help you understand but they are somewhat similar.
1. 상태: condition, state
ex1) 나는 오늘 몸 상태가 안 좋다. = My physical condition is poor today. = I don't feel good today.
ex2) 이 자동차의 상태가 나쁘다 = This car's condition is bad.
2.사태: event, situation
ex1) 우리는 비상사태를 위해 대기중입니다. = We are on standby in case of an emergency.
ex2) 산사태 = landslide (mountain '산' + event '사태')
by the way, 사태 is a homonym(동음이의어) and it also means a part of beef.
3.상황: situation, circumstance
ex1) 상황에 따라 필요한 물건들이 달라집니다. = Depending on the situation, things we need can vary.
ex2) 긴급상황에 대비하는 것이 필수 입니다. = It is necessary to be prepared for an emergency.
Hope it helps!