Spinal cord tumors significantly contribute to neurological morbidity and mortality. Precise morphometric quantification, encompassing the size, location, and type of such tumors, holds promise for optimizing treatment planning strategies. Although recent methods have demonstrated excellent performance in medical image segmentation, they primarily focus on discerning shapes with relatively large morphology such as brain tumors, ignoring the challenging problem of identifying spinal cord tumors which tend to have tiny sizes, diverse locations, and shapes. To tackle this hard problem of multiclass spinal cord tumor segmentation, we propose a new method, called BATseg, to learn a tumor surface distance field by applying our new multiclass boundary-aware loss function. To verify the effectiveness of our approach, we also introduce the first and large-scale spinal cord tumor dataset. It comprises gadolinium-enhanced T1-weighted 3D MRI scans from 653 patients and contains the four most common spinal cord tumor types: astrocytomas, ependymomas, hemangioblastomas, and spinal meningiomas. Extensive experiments on our dataset and another public kidney tumor segmentation dataset show that our proposed method achieves superior performance for multiclass tumor segmentation.