In this paper, we present a comprehensive UAV system design to perform the highly complex task of off-centered aerial grasping. This task has several interdisciplinary research challenges which need to be addressed at once. The main design challenges are GPS-denied functionality, solely onboard computing, and avoiding off-the-shelf costly positioning systems. While in terms of algorithms, visual perception, localization, control, and grasping are the leading research problems. Hence in this paper, we make interdisciplinary contributions: (i) A detailed description of the fundamental challenges in indoor aerial grasping, (ii) a novel lightweight gripper design, (iii) a complete aerial platform design and in-lab fabrication, and (iv) localization, perception, control, grasping systems, and an end-to-end flight autonomy state-machine. Finally, we demonstrate the resulting aerial grasping system Drone-Bee achieving a high grasping rate for a highly challenging agricultural task of apple-like fruit harvesting, indoors in a vertical farming setting (Fig. 1). To our knowledge, such a system has not been previously discussed in the literature, and with its capabilities, this system pushes aerial manipulation towards 4th generation.