Cancer patients tend to have a high psychological burden. Half of cancer patients suffer from severe affective disorders and anxiety disorders, while one-third struggle with mild forms of these. The COVID-19 pandemic is damaging the mental health of the population due to social restrictions. A growing number of studies note the role of COVID-19 anxiety in the health and quality of life of cancer patients. The purpose of this study is to estimate the level of COVID-19 anxiety among oncology patients and to test the utility of the FCV-19S scale in a population study of cancer patients. The study included 600 respondents (300 oncology patients and 300 control subjects not undergoing oncological treatment). The FCV-19S scale and the GAD-7 scale were used in the study. The results were interpreted according to the following verbal scale: 76–100%, high anxiety; 56–75%, moderate anxiety; 26–55%, low COVID-19 anxiety; <25%, no COVID-19 anxiety. In the analysis of the GAD-7 questionnaire results, the mean score obtained was 8.21 (min. 0; max. 21; SD 5.32). For 81% of respondents in the group of oncology patients, the total score indicated the presence of anxiety symptoms with varying degrees of severity; in the control group, this proportion was 55% of respondents. The FCV-19S scale score as a percentage was 57.4% for oncology patients, indicating a moderate level of fear of the SARS-CoV-2 virus, and 30.3% for the control group, indicating a low level of fear of the SARS-CoV-2 virus. One-fifth of oncology patients were afraid of losing their lives due to the SARS-CoV-2 virus; in the control group, this proportion was 13% of respondents. Oncology patients were characterized by a higher prevalence of sleep disturbance than control group respondents, which was associated with greater anxiety. The study, therefore, shows that oncology patients have moderate levels of anxiety associated with the COVID-19 pandemic, and non-oncology patients show lower levels of anxiety.