3D instance segmentation is fundamental to geometric understanding of the world around us. Existing methods for instance segmentation of 3D scenes rely on supervision from expensive, manual 3D annotations.We propose UnScene3D, the first fully unsupervised 3D learning approach for class-agnostic 3D instance segmentation of indoor scans. UnScene3D first generates pseudo masks by leveraging self-supervised color and geometry features to find potential object regions. We operate on a basis of 3D segment primitives, enabling efficient representation and learning on high-resolution 3D data. The coarse proposals are then refined through self-training our model on its predictions. Our approach improves over state-of-the-art unsupervised 3D instance segmentation methods by more than 300% Average Precision score, demonstrating effective instance segmentation even in challenging, cluttered 3D scenes.