Our paper focuses on the issues of social health and psychological safety of university students involved in digital sustainable education during the COVID-19 pandemic. Currently, modern education is becoming inclusive due to the advancements in information and communication technologies (ICT), and it is important not only to stress the relevance of sustainable development and the use of digital technologies, but also their impact on students at schools and universities worldwide. Digital literacy is a newly emerging feature that results from the attitude of team members in the field of digital technologies. This paper explores the impacts of the COVID-19 pandemic on students’ learning and well-being and outlines the potential considerations for educational systems as they support students through the recovery period and beyond. Our study is based on the results of our own survey that was administered using a snowball and convenient sample of 1524 respondents (aged 19–26 years; 56.2% females and 43.8% males) from the Czech Republic (N = 804) and Russia (N = 720). We employed the ANOVA and Dirichlet Process mixtures of Generalized Linear Models (DP-GLM) in order to explain the causes of stress and anxiety after grouping variables represented by gender and the study specializations. Our results demonstrate that more than 87% of the students in the sample expressed a medium to high vulnerability to stress, while 58% of the respondents were affected by severe anxiety during their online education engagement. The most important factors that emerged as significant were the fear of getting infected and social distancing, while the best strategy to cope with the stress was self-control. These results allow us to provide practical recommendations for effectively coping with and controlling stress and anxiety among students in the post-pandemic era. In addition, our findings might contribute considerably to the study of the overall long-term effect of the COVID-19 pandemic on the university students, in general, and the use of digital technologies in higher education, as well as on the public health.