Remote monitoring of vital signs for studying sleep is a user-friendly alternative to monitoring with sensors attached to the skin. For instance, remote monitoring can allow unconstrained movement during sleep, whereas detectors requiring a physical contact may detach and interrupt the measurement and affect sleep itself. This study evaluates the performance of a cost-effective frequency modulated continuous wave (FMCW) radar in remote monitoring of heart rate and respiration in scenarios resembling a set of normal and abnormal physiological conditions during sleep. We evaluate the vital signs of ten subjects in different lying positions during various tasks. Specifically, we aim for a broad range of both heart and respiration rates to replicate various real-life scenarios and to test the robustness of the selected vital sign extraction methods consisting of fast Fourier transform based cepstral and autocorrelation analyses. As compared to the reference signals obtained using Embla titanium, a certified medical device, we achieved an overall relative mean absolute error of $3.6$% (86% correlation) and $9.1$% (91% correlation) for the heart rate and respiration rate, respectively. Our results promote radar-based clinical monitoring by showing that the proposed radar technology and signal processing methods accurately capture even such alarming vital signs as minimal respiration. Furthermore, we show that common parameters for heart rate variability can also be accurately extracted from the radar signal, enabling further sleep analyses.