Radar-based human activity recognition (HAR) is attractive for unobtrusive and privacy-preserving monitoring, yet many CNN/RNN solutions remain too heavy for edge deployment, and even lightweight ViT/SSM variants often exceed practical compute and memory budgets. We introduce Neural-HAR, a dimension-gated CNN accelerator tailored for real-time radar HAR on resource-constrained platforms. At its core is GateCNN, a parameter-efficient Doppler-temporal network that (i) embeds Doppler vectors to emphasize frequency evolution over time and (ii) applies dual-path gated convolutions that modulate Doppler-aware content features with temporal gates, complemented by a residual path for stable training. On the University of Glasgow UoG2020 continuous radar dataset, GateCNN attains 86.4% accuracy with only 2.7k parameters and 0.28M FLOPs per inference, comparable to CNN-BiGRU at a fraction of the complexity. Our FPGA prototype on Xilinx Zynq-7000 Z-7007S reaches 107.5 $μ$s latency and 15 mW dynamic power using LUT-based ROM and distributed RAM only (zero DSP/BRAM), demonstrating real-time, energy-efficient edge inference. Code and HLS conversion scripts are available at https://github.com/lab-emi/AIRHAR.