The ongoing intense development of short-range radar systems and their improved capability of measuring small movements make these systems reliable solutions for the extraction of human vital signs in a contactless fashion. The continuous contactless monitoring of vital signs can be considered in a wide range of applications, such as remote healthcare solutions and context-aware smart sensor development. Currently, the provision of radar-recorded datasets of human vital signs is still an open issue. In this paper, we present a new frequency-modulated continuous wave (FMCW) radar-recorded vital sign dataset for 50 children aged less than 13 years. A clinically approved vital sign monitoring sensor was also deployed as a reference, and data from both sensors were time-synchronized. With the presented dataset, a new child age-group classification system based on GoogLeNet is proposed to develop a child safety sensor for smart vehicles. The radar-recorded vital signs of children are divided into several age groups, and the GoogLeNet framework is trained to predict the age of unknown human test subjects.