(1) Background: The issue of burnout in healthcare staff is frequently discussed in relation to occupational health. In this paper, we report healthcare staff experiences of stress and burnout. (2) Methods: In total, 72 healthcare staff were interviewed from psychiatry, surgery, and emergency departments at an Australian public health service. The sample included doctors, nurses, allied health professionals, administrators, and front-line managers. Interview transcripts were thematically analyzed, with participant experiences interpreted against descriptors of burnout in Maslach’s Burnout Inventory and the International Statistical Classification of Diseases and Related Health Problems (ICD-11). (3) Results: Staff experiences closely matched the ICD-11 description of stress associated with working in an uncongenial workplace, with few reported experiences which matched the ICD-11 descriptors of burnout. (4) Conclusion: Uncongenial workplaces in public health services contribute to healthcare staff stress. While previous approaches have focused on biomedical assistance for individuals, our findings suggest that occupational health approaches to addressing health care staff stress need greater focus on the workplace as a social determinant of health. This finding is significant as organizational remedies to uncongenial stress are quite different from remedies to burnout.