Deep learning-based lossless compression methods offer substantial advantages in compressing medical volumetric images. Nevertheless, many learning-based algorithms encounter a trade-off between practicality and compression performance. This paper introduces a hardware-friendly streaming lossless volumetric compression framework, utilizing merely one-thousandth of the model weights compared to other learning-based compression frameworks. We propose a gated recurrent convolutional neural network that combines diverse convolutional structures and fusion gate mechanisms to capture the inter-slice dependencies in volumetric images. Based on such contextual information, we can predict the pixel-by-pixel distribution for entropy coding. Guided by hardware/software co-design principles, we implement the proposed framework on Field Programmable Gate Array to achieve enhanced real-time performance. Extensive experimental results indicate that our method outperforms traditional lossless volumetric compressors and state-of-the-art learning-based lossless compression methods across various medical image benchmarks. Additionally, our method exhibits robust generalization ability and competitive compression speed