Unsupervised cross-modal hashing has gained increasing attention for its ability to efficiently retrieve semantically relevant data across different modalities without requiring labeled supervision. However, existing methods often suffer from limited feature representation and insufficient semantic alignment due to their reliance on shallow extractors and coarse global features. To address these limitations, we propose a novel method called MGCH (Multi-Granularity feature and Contrastive learning Hashing), which integrates multi-granularity feature learning with contrastive learning under an unsupervised framework. Specifically, MGCH adopts CLIP to extract both global and local features from images and texts. An intra-modal multi-granularity module guides local feature learning via global semantics to enhance fine-grained representation. Furthermore, an inter-modal contrastive learning module is designed using a fusion strategy and Transformer encoder to facilitate deep semantic interaction across modalities. Finally, compact binary hash codes are generated from both intra-modal and inter-modal features for efficient retrieval. Extensive experiments on two benchmark datasets, MIRFlickr-25K and NUS-WIDE, demonstrate that MGCH achieves superior performance over state-of-the-art methods.