Background The COVID-19 pandemic is currently one of the biggest public health threats for people’s mental health. A particularly endangered group were students, who became highly affected by measures of social distance due to their active lifestyle. Therefore, the aim of this paper is to assess the level of self-reported stress, anxiety and depression of the student population in Serbia, in relation to demographic characteristics, living and studying conditions, students’ activities during the epidemic, potential coronavirus infection and general, mental and physical health. Methods We conducted a cross-sectional study of 580 undergraduate medical students from the University of Belgrade during the 2020/2021 school year. Mental health data were collected using the Depression Anxiety Stress Scales (DASS-21). Both bivariate and multivariate logistic regression analysis were used to examine the association between independent variables with the dependent variable mental health. Results Women made up the majority of the sample with 80.3%. A total of 64.5%, 66.8% and 66.7% of students between the ages of 21 and 30 had severe depressive symptoms, severe degree of anxiety, and a severe degree of stress, respectively. Women almost twice as often (OR = 1.89) assessed their anxiety as severe and almost two and a half times more (OR = 2.39) perceived stress as severe compared to men. Students who lived with their families during studies two and a half times (OR = 2.57) more often assessed their stress as severe, compared to students who lived alone. Fifth- and sixth-year students were less likely to rate depression and anxiety as serious than the first-year students. Conclusions Medical students reported their health as severely impaired in terms of depression, anxiety and stress reactions. The results indicate the need to launch a mental health program in the form of counseling and emotional support to students affected by the pandemic.