The emergence of the novel coronavirus in December 2019 in China marked the beginning of a pandemic that impacted healthcare systems and economic life all over the world. The virus primarily targets the respiratory system causing severe acute respiratory syndrome (SARS) in some patients, and therefore received the name of SARS-CoV-2. The pathogen stands out among other coronaviruses by its rapid transmission from human to human, with the majority of infected individuals being asymptomatic or presenting with only minor illness, therefore facilitating the pathogen spread. At the same time, people from the risk groups, such as the elderly, patients suffering from chronic diseases, or obese individuals, have increased chances of developing a severe or even fatal disease. The search for risk factors explaining this phenomenon continues. In this review, we focus on the known mechanisms of SARS-CoV-2 infection affecting the functioning of the immune system and discuss potential risk factors responsible for the severe disease course. Oxidative stress is one of such factors, which plays a prominent role in innate immunity activity, and recent research has revealed its tight involvement in SARS-CoV-2 infection. We discuss these recent findings and the development of excessive inflammation and cytokine storm observed during SARS-CoV-2 infection. Finally, we consider potential use of antioxidant drugs for alleviating the severe symptoms in affected patients.