Background: Mental health is an important issue for students during the time of the COVID-19 pandemic and uncertainty. Students suffer from mental health issues due to delayed academic years and prolong stay at home during the lockdown. This study aimed to identify factors associated with depression, anxiety, and stress among undergraduate health sciences students of different medical institutions in Nepal. Materials and Methods: A web-based cross-sectional survey was conducted among 493 health sciences students between 14 July and 16 August 2020. Depression, anxiety, and stress were measured using the Depression, Anxiety, Stress Scale-21 (DASS-21). Multivariable logistic regression analysis was performed to determine the risk factors of mental health outcomes. Results: Overall, 50.5, 52.5, and 44.6% of students had symptoms of depression, anxiety, and stress, respectively. Significantly higher odds of experiencing stress symptoms [adjusted odds ratio (AOR): 2.166; 95% CI: 1.075–4.363] were found with the participants whose relatives were infected with COVID-19. Participants with age less than or equal to 21 years among undergraduate health sciences students were significantly associated with higher odds of experiencing symptoms of stress (AOR: 1.626; 95% CI: 1.110–2.383) and anxiety (AOR: 1.6251; 95% CI: 1.110–2.379) in comparison with age above 21. Staying in quarantine was significantly associated with higher odds of experiencing depressive symptoms (AOR: 2.175; 95% CI: 1.142–4.143). Participants who had internet facilities at the residence had less likely to have depressive symptoms than those who are lacking internet services (AOR: 0.420; 95% CI: 0.195–0.905). Conclusions: Staying in quarantine had higher odds of having depression and students who had internet facilities had lower odds of having depression. While staying in quarantine or isolation, it would be better to provide things to engage like the internet. A focus on improving the mental well-being of health sciences students should be initiated immediately after such a pandemic and lockdown.