Medical professions are characterized by a great responsibility for human health and life; they are also vulnerable to burnout. The outbreak of the COVID-19 pandemic has brought new challenges and threats. This study aimed to assess the mental health of healthcare workers after a year and a half of working in COVID-19 pandemic conditions. The Perceived Stress Scale (PSS-10), the Link Burnout Questionnaire (LBQ), and the General Health Questionnaire (GHQ-28) were utilized in this cross-sectional investigation. A total of 335 healthcare employees from Polish hospitals (median age 44 years) filled out online questionnaires between the 16 August 2021 and the 30 March 2022. Most of the sample was female (86%). In this study, 40.0% of the surveyed healthcare workers reported a high stress intensity. Burnout was reported by 9.6% of the workers, and the most frequently experienced symptom was psychophysical exhaustion. Almost half of the healthcare workers surveyed (49.6%) reported health disorders at both the mental and physiological levels. Interestingly, working in a COVID-19 ward did not significantly differentiate healthcare workers in any of the evaluated variables: PSS-10 (gr. A F = 1.21; gr. B F = 0.71; p > 0.05), LBQ (gr. A F = 1.89, F = 0.94, F = 1.08, F = 2.57; gr. B F = 0.32, F = 1.14, F = 0.77, F = 0.36; p > 0.05), and GHQ-28 (gr. A F = 0.85, F = 0.52, F = 0.57, F = 0.31; gr. B F = 0.31, F = 0.06, F = 0.06, F = 0.54; p > 0.05). Furthermore, there were no statistically significant differences between the compared occupational groups of healthcare workers: PSS-10 (F = 1.08; p > 0.05) and GHQ-28 (F = 1.78; F = 0.85; F = 0.62; F = 0.54; p > 0.05). The mental health of healthcare workers is alarming, and psychophysical conditions can affect the quality of work and relations with patients. Psychological care in workplaces and workshops that build resources for dealing with difficult situations are necessary.