Dwarf spheroidal galaxies are excellent systems to probe the nature of fermionic dark matter due to their high observed dark matter phase-space density. In this work, we review, revise, and improve upon previous phase-space considerations to obtain lower bounds on the mass of fermionic dark matter particles. The refinement in the results compared to previous works is realized particularly due to a significantly improved Jeans analysis of the galaxies. We discuss two methods to obtain phase-space bounds on the dark matter mass, one model-independent bound based on Pauli's principle, and the other derived from an application of Liouville's theorem. As benchmark examples for the latter case, we derive constraints for thermally decoupled particles and (non-)resonantly produced sterile neutrinos. Using the Pauli principle, we report a model-independent lower bound of m >= 0.18 keV at 68 per cent CL and m >= 0.13 keV at 95 per cent CL. For relativistically decoupled thermal relics, this bound is strengthened to m >= 0.59 keV at 68 per cent CL and m >= 0.41 keV at 95 per cent CL, while for non-resonantly produced sterile neutrinos the constraint is m >= 2.80 keV at 68 per cent CL and m >= 1.74 keV at 95 per cent CL. Finally, the phase-space bounds on resonantly produced sterile neutrinos are compared with complementary limits from X-ray, Lyman alpha, and big bang nucleosynthesis observations.