The deep gap states created by defects in semiconductors typically deteriorate the performance of (opto)electronic devices. This has limited the applications of two-dimensional (2D) metal dichalcogenides (MX2) and underscored the need for a new 2D semiconductor without defect-induced deep gap states. In this work, we demonstrate that a 2D mono-elemental semiconductor is a promising candidate. This is exemplified by first-principles study of 2D phosphorus (P), a recently fabricated high-mobility semiconductor. Most of the defects, including intrinsic point defects and grain boundaries, are electronically inactive, thanks to the homoelemental bonding, which is not preferred in heteroelemental system such as MX2. Unlike MX2, the edges of which create deep gap states and cannot be eliminated by passivation, the edge states of 2D P can be removed from the band gap by hydrogen termination. We further find that both the type and the concentration of charge carriers in 2D P can be tuned by doping with foreign atoms. Our work sheds light on the role of defects in the electronic structure of materials.