The COVID-19 pandemic imposed not only serious threats to the physical health of the population, but also provoked a wide range of psychological problems. The study was aimed to define the structure of anxiety in the population during the epidemic period, as well as to identify the most vulnerable social groups (including individuals with affective disorders) which were most in need of psychological and/or psychiatric help. The online survey of 1957 Russian-speaking respondents aged over 18 was carried out from March 30 to April 5, 2020. The anxiety distress level was verified using the Psychological Stress Measure (PSM-25), the stigmatization of individuals experiencing respiratory symptoms was assessed using the modified Perceived Devaluation-Discrimination Questionnaire (PDD; Cronbach's α = 0.707). In 99.8% of respondents, the combination of various concerns associated with COVID-19 was observed, the mean psychological stress score was increased to moderate level (score 104.9 ± 34.4), and the stigmatization score exceeded the whole sample median value (19.5±3.4; Me = 17). About 35% of respondents had concerns associated with anxiety distress (Cohen’s d = 0.16–0.39): these were the "risk of social isolation" and the "possible lack of medication for daily use". The following groups of respondents were the most susceptible to the stress: people with affective disorders, young people (aged ≤20), unemployed persons, single persons, people with no formal education, and women. Thus, the broad sectors of the population need correction of anxiety distress associated with the COVID-19 pandemic. Therefore, the measures’ implementation should be targeted, and in terms of coverage and content oriented to the identified vulnerable social groups.