The integration of complementary characteristics from camera and radar data has emerged as an effective approach in 3D object detection. However, such fusion-based methods remain unexplored for place recognition, an equally important task for autonomous systems. Given that place recognition relies on the similarity between a query scene and the corresponding candidate scene, the stationary background of a scene is expected to play a crucial role in the task. As such, current well-designed camera-radar fusion methods for 3D object detection can hardly take effect in place recognition because they mainly focus on dynamic foreground objects. In this paper, a background-attentive camera-radar fusion-based method, named CRPlace, is proposed to generate background-attentive global descriptors from multi-view images and radar point clouds for accurate place recognition. To extract stationary background features effectively, we design an adaptive module that generates the background-attentive mask by utilizing the camera BEV feature and radar dynamic points. With the guidance of a background mask, we devise a bidirectional cross-attention-based spatial fusion strategy to facilitate comprehensive spatial interaction between the background information of the camera BEV feature and the radar BEV feature. As the first camera-radar fusion-based place recognition network, CRPlace has been evaluated thoroughly on the nuScenes dataset. The results show that our algorithm outperforms a variety of baseline methods across a comprehensive set of metrics (recall@1 reaches 91.2%).