Free-Viewpoint Video (FVV) reconstruction enables photorealistic and interactive 3D scene visualization; however, real-time streaming is often bottlenecked by sparse-view inputs, prohibitive training costs, and bandwidth constraints. While recent 3D Gaussian Splatting (3DGS) has advanced FVV due to its superior rendering speed, Streaming Free-Viewpoint Video (SFVV) introduces additional demands for rapid optimization, high-fidelity reconstruction under sparse constraints, and minimal storage footprints. To bridge this gap, we propose StreamLoD-GS, an LoD-based Gaussian Splatting framework designed specifically for SFVV. Our approach integrates three core innovations: 1) an Anchor- and Octree-based LoD-structured 3DGS with a hierarchical Gaussian dropout technique to ensure efficient and stable optimization while maintaining high-quality rendering; 2) a GMM-based motion partitioning mechanism that separates dynamic and static content, refining dynamic regions while preserving background stability; and 3) a quantized residual refinement framework that significantly reduces storage requirements without compromising visual fidelity. Extensive experiments demonstrate that StreamLoD-GS achieves competitive or state-of-the-art performance in terms of quality, efficiency, and storage.