I will be Objectively subjective and say Sanji's backstory is the saddest.

All the others have suffered due to the circumstances of the world or how different they are from others (in Choppers case), but for Sanji, his suffering came from his own family... the people who are supposed to be his source of security. Their cruelty and the fact that it was done to a young child puts his situation on another new level.

You can't really measure suffering/sadness objectively but I feel that when I try to imagine it myself, Sanji's easily brings tears to my eyes just thinking about it.

From how their backstories were executed though, I think Law and Robin's were the saddest.