Prosthetic hands, while having shown significant progress in affordability, typically suffer from limited repairability, specifically by the user themselves. Several modular hands have been proposed to address this, but these solutions require handling of intricate components or are unsuitable for prosthetic use due to the large volume and weight resulting from added mechanical complexity to achieve this modularity. In this paper, we propose a fully modular design for a prosthetic hand with finger and wrist level modularity, allowing the removal and attachment of tendon-driven fingers without the need for tools, retendoning, and rewiring. Our innovative design enables placement of the motors behind the hand for remote actuation of the tendons, which are contained solely within the fingers. Details of the novel coupling-transmission mechanisms enabling this are presented; and the capabilities of a prototype using a control-independent grasping benchmark are discussed. The modular detachment torque of the fingers is also computed to analyse the trade-off between intentional removal and the ability to withstand external loads. Experiment results demonstrate that the prosthetic hand is able to grasp a wide range of household and food items, of different shape, size, and weight, without resulting in the ejection of fingers, while allowing a user to remove them easily using a single hand.