How sound is the story that talks about Abu Bakr's appointment of Umar as a judge over Medina, and Umar's staying for a year without two people disputing before him, and his request to be relieved from the judiciary, explaining that the reason for his request was due to the lack of need for him among a believing people, and that they knew what was for them and what was against them, and each of them loved for his brother what he loved for himself?

Firstly: It has been reported that Umar remained for a year without judging between two people when Abu Bakr appointed him to the judiciary. However, the chain of transmission for this narration is not sound and is weak, because it consists of maraseel (pl. of mursal - a narration in which a Tabi'i narrates directly from the Prophet without mentioning the Companion), and none of them are authentic. Thus, the report is weak and is not strengthened by multiple chains of transmission.

Secondly: The story mentioned in the question with its details (Umar's resignation from the judiciary because Muslims had no need for him) has no basis. Muslims are always in need of a judge, and disputes occurred even during the time of the Prophet, peace be upon him.