Background: Since the outbreak of the COVID-19 pandemic, healthcare workers (HCWs) have been faced with specific stressors endangering their physical and mental health and their functioning. This study aimed to assess the short-term psychological health of a sample of Italian HCWs and the related influencing factors. In particular, the study focused on the differences related to HCWs’ gender and to having been directly in charge of COVID-19 patients or not. Methods: An online survey was administered to the whole staff of the Modena General University Hospital three months after the onset of the pandemic, in 2020. Demographic data and changes in working and living conditions related to COVID-19 were collected; mental health status was assessed by the Depression, Anxiety and Stress Scale (DASS-21) and the Impact of Event Scale-Revised (IES-R). Results: 1172 out of 4788 members returned the survey (response rate = 24.5%), the male/female ratio was 30/70%. Clinically significant symptoms assessed according to the DASS-21 emerged among 21.0% of the respondents for depression, 22.5% for anxiety and 27.0% for stress. Symptoms suggestive of a traumatic reaction were reported by 19.0% of the sample. Symptoms of psychological distress were statistically associated with female gender, job role, ward, changes in lifestyle, whereas first-line work with COVID-19 patients was statistically associated with more stress symptoms. HCWs reported a significant level of psychological distress that could reach severe clinical significance and impact dramatically their quality of life and functioning. Conclusions: Considering the persistence of the international emergency, effective strategies to anticipate, recognize and address distress in HCWs are essential, also because they may impact the organization and effectiveness of healthcare systems.