A sob story is any information in a post other than a description of what is wanted, possibly considered to be a way of getting the sympathy of the reader and making them more inclined to provide the poster with the item they are requesting.
Beyond that the definition is very subjective. Members may find excessive detail time consuming and possibly choose to unsubscribe as a result. On the other hand, this sort of narrative can make reading a wanted post more interesting. There is no way of verifying sob stories. Some groups do not allow sob stories, others approve posts leaving such judgements to the discretion of their members.
Another consideration is that there is also a possible safety risk in giving too much personal detail in a post. It may be appropriate to edit or even return a post that includes too much personal info or is a 'sob story', but it's important that this is done prudently and with much sensitivity.