Creating detailed 3D characters from a single image remains challenging due to the difficulty in separating semantic components during generation. Existing methods often produce entangled meshes with poor topology, hindering downstream applications like rigging and animation. We introduce SeparateGen, a novel framework that generates high-quality 3D characters by explicitly reconstructing them as distinct semantic components (e.g., body, clothing, hair, shoes) from a single, arbitrary-pose image. SeparateGen first leverages a multi-view diffusion model to generate consistent multi-view images in a canonical A-pose. Then, a novel component-aware reconstruction model, SC-LRM, conditioned on these multi-view images, adaptively decomposes and reconstructs each component with high fidelity. To train and evaluate SeparateGen, we contribute SC-Anime, the first large-scale dataset of 7,580 anime-style 3D characters with detailed component-level annotations. Extensive experiments demonstrate that SeparateGen significantly outperforms state-of-the-art methods in both reconstruction quality and multi-view consistency. Furthermore, our component-based approach effectively resolves mesh entanglement issues, enabling seamless rigging and asset reuse. SeparateGen thus represents a step towards generating high-quality, application-ready 3D characters from a single image. The SC-Anime dataset and our code will be publicly released.