Cyclists are considered to be vulnerable road users (VRUs) and need protection from potential collisions with cars and other vehicles induced by unsafe driving, dangerous road conditions, or weak cycling infrastructure. Integrating mmWave radars into cycling safety measures presents an efficient solution to this problem given their compact size, low power consumption, and low cost compared to other sensors. This paper introduces an mmWave radar-based bike safety system designed to offer real-time alerts to cyclists. The system consists of a low-power radar sensor affixed to the bicycle, connected to a micro-controller, and delivering a preliminary classification of detected obstacles. An efficient two-level clustering based on the accumulation of radar point clouds from multiple frames with a temporal projection from previous frames into the current frame is proposed. The clustering is followed by a coarse classification algorithm in which we use relevant features extracted from the resulting clusters. An annotated RadBike dataset composed of radar point cloud data synchronized with RGB camera images is developed to evaluate our system. The two-level clustering outperforms the DBSCAN algorithm, achieving a v-measure score of 0.91, compared to 0.88 with classical DBSCAN. Different classifiers, including decision trees, random forests, support vector machines (SVMs), and AdaBoost, have been assessed, with an overall accuracy of 87% for the three main object classes: four-wheeled, two-wheeled, and others. The system has the ability to improve rider safety on the road and substantially reduce the frequency of incidents involving cyclists.