Background This study investigates the relationship between exposure to stress, trait emotional intelligence (trait EI) and mental health problems (anxiety and depression symptoms) amongst basic-year medical students during COVID-19. The sample consists of 379 basic-year medical students. Off them, 160 (42.4%) were male and 218 (57.5%) were female. The following measures were used in the study: The Medical Student Stressor Questionnaire (MSSQ), Trait Emotional Intelligence Questionnaire—Short Form (TEIQue-ASF), The anxiety symptoms scale and the depression scale. Results The majority of students reported high to severe level ARS and mild to moderate DRS. Female students showed higher levels of ARS, TLRS, SRS, GARS and anxiety symptoms compared to male ones. Moreover, students with higher levels of academic performance reported lower levels of ARS and DRS, higher levels of trait EI and lower levels of anxiety and depression symptoms. In addition, trait EI has a significant negative association with anxiety and depression symptoms and stress domains (ARS, IRS, TLRS, SRS, DRS, GARS) and total stress. Finally, stress domains are positively correlated with anxiety and depression symptoms. Conclusion Basic-year medical students in the Gaza Strip are exposed to stressful events which aggravate the effect of exposure and cause higher levels of anxiety and depression symptoms.