Three-dimensional (3D) point clouds are increasingly used in applications such as autonomous driving, robotics, and virtual reality (VR). Point-based neural networks (PNNs) have demonstrated strong performance in point cloud analysis, originally targeting small-scale inputs. However, as PNNs evolve to process large-scale point clouds with hundreds of thousands of points, all-to-all computation and global memory access in point cloud processing introduce substantial overhead, causing$O\left(n^{2}\right)$computational complexity and memory traffic where$n$is the number of points. Existing accelerators, primarily optimized for small-scale workloads, overlook this challenge and scale poorly due to inefficient partitioning and non-parallel architectures. To address these issues, we propose FractalCloud, a fractal-inspired hardware architecture for efficient large-scale 3D point cloud processing. FractalCloud introduces two key optimizations: (1) a co-designed Fractal method for shape-aware and hardware-friendly partitioning, and (2) block-parallel point operations that decompose and parallelize all point operations. A dedicated hardware design with on-chip fractal and flexible parallelism further enables fully parallel processing within limited memory resources. Implemented in 28 nm technology as a chip layout with a core area of$1.5 ~\text{mm}^{2}$, FractalCloud achieves$21.7 \times$speedup and$27 \times$energy reduction over state-of-the-art accelerators while maintaining network accuracy, demonstrating its scalability and efficiency for PNN inference. The code for FractalCloud is available at https://github.com/Yuzhe-Fu/FractalCloud.