Background/Objectives: The most common post-acute consequences of SARS-CoV-2 include lung dysfunction, the impairment of cognitive functions and mental health, as well as the impairment of the musculoskeletal system in the form of fatigue and muscle weakness. Post-COVID-19 patients often experience impaired balance and reduced physical capacity. It is important to implement a rehabilitation program that eliminates the side effects of COVID-19 and allows for significant improvement in the patient’s functionality. The aim of our study was to assess patient functionality after a 6-week rehabilitation program on balance, foot pressure distribution, and physical capacity in post-COVID-19 patients. Methods: The clinical study group consisted of 53 people 3 months after COVID-19 infection, confirmed by a positive PCR test. Exclusion from the study included people with comorbidities that impaired balance and gait. The patients underwent a posturographic assessment—Romberg test, a baropodometric assessment—static and dynamic, and a performance assessment—a 6 min walk test determining shortness of breath on the mMRC scale, blood pressure, heart rate, and saturation. Patients participated in rehabilitation until the sixth week, after which they were assessed again. Comparisons were made using IBM SPSS Statistics 27.0 software using the Wilcoxon pairwise order test, at a significance level of p < 0.05. Results: The result of the postural control assessment showed an improvement in the ability to maintain the centre of gravity in terms of the foot support area—statistical decreases were observed in the ellipse area, from 745.28 mm2 to 453.52 mm2 (p = 0.009), as well as maximum (from 3133.5 gr/cm2 to 2994.2 gr/cm2; p = 0.065) and average load on the left foot (from 1010.1 gr/cm2 to 969.38 gr/cm2; p = 0.028). In the 6 min walk test before and after exercise, the heart rate decreased after the therapy (shortness of breath on the mMRC scale also decreased from 79.12 to 74.95). This means that patients achieved better physical fitness and efficiency. Conclusions: Rehabilitation significantly improved balance, as measured by a decrease in ellipse area during the Romberg test.