The 24 GHz narrowband frequency-modulated continuous-wave (FMCW) radar has been widely applied in automotive and other vital sensors. However, its range resolution depends on the bandwidth, which is limited to 60 cm in its narrowband of 24.0-24.25 GHz. Furthermore, at low-range resolutions, the power spectral density (PSD) of the fast Fourier transform and multiple signal classification algorithms typically used for distance measurements often suffer from sidelobe interference between the signals. In particular, under low signal-to-noise ratios (SNRs), the distance-estimation errors significantly increases. This is a major obstacle in the development of applications that requiring high accuracy, such as autonomous driving and patient-health monitoring. Therefore, we propose a technique to enhance the range resolution and noise resistance of the 24 GHz FMCW radar using nonharmonic analysis (NHA), a high-resolution frequency-analysis method. In the PSD of NHA, a sinusoidal signal is represented as a single-line spectrum. Experimental results using a metal reflector demonstrate that the proposed method significantly suppresses sidelobes and achieves a high range resolution, as evidenced by the sparse PSDs obtained. Furthermore, simulations and a pedestrian-tracking experiment show that the proposed method achieves lower RMSE and higher peak SNR than conventional FFT and MUSIC methods, even at low input SNRs. This method is expected to enhance the performance of various radar applications, such as vehicle-perimeter monitoring, robot navigation and indoor monitoring.