Digital humans have witnessed extensive applications in various domains, necessitating related quality assessment studies. However, there is a lack of comprehensive digital human quality assessment (DHQA) databases. To address this gap, we propose SJTU-H3D, a subjective quality assessment database specifically designed for full-body digital humans. It comprises 40 high-quality reference digital humans and 1,120 labeled distorted counterparts generated with seven types of distortions. The SJTU-H3D database can serve as a benchmark for DHQA research, allowing evaluation and refinement of processing algorithms. Further, we propose a zero-shot DHQA approach that focuses on no-reference (NR) scenarios to ensure generalization capabilities while mitigating database bias. Our method leverages semantic and distortion features extracted from projections, as well as geometry features derived from the mesh structure of digital humans. Specifically, we employ the Contrastive Language-Image Pre-training (CLIP) model to measure semantic affinity and incorporate the Naturalness Image Quality Evaluator (NIQE) model to capture low-level distortion information. Additionally, we utilize dihedral angles as geometry descriptors to extract mesh features. By aggregating these measures, we introduce the Digital Human Quality Index (DHQI), which demonstrates significant improvements in zero-shot performance. The DHQI can also serve as a robust baseline for DHQA tasks, facilitating advancements in the field. The database and the code are available at https://github.com/zzc-1998/SJTU-H3D.