We present NeCGS, the first neural compression paradigm, which can compress a geometry set encompassing thousands of detailed and diverse 3D mesh models by up to 900 times with high accuracy and preservation of detailed geometric structures. Specifically, we first propose TSDF-Def, a new implicit representation that is capable of accurately representing irregular 3D mesh models with various structures into regular 4D tensors of uniform and compact size, where 3D surfaces can be extracted through the deformable marching cubes. Then we construct a quantization-aware auto-decoder network architecture to regress these 4D tensors to explore the local geometric similarity within each shape and across different shapes for redundancy removal, resulting in more compact representations, including an embedded feature of a smaller size associated with each 3D model and a network parameter shared by all models. We finally encode the resulting features and network parameters into bitstreams through entropy coding. Besides, our NeCGS can handle the dynamic scenario well, where new 3D models are constantly added to a compressed set. Extensive experiments and ablation studies demonstrate the significant advantages of our NeCGS over state-of-the-art methods both quantitatively and qualitatively. We have included the source code in the Supplemental Material.