Quantizing to <= 256 colors will let you use a single byte per pixel, but there are other techniques like Block Truncation Coding that work well with 8bit images to go down to 2 bits per pixel or lower. Even at 2 bits per pixel, this is still quite big as raw data, so you typically will want to use compression on top such as RLE, DEFLATE, etc.

I’m currently exploring this for my own applications, compressing large 8bit sprite sheet images, and it’s producing much smaller images than most palette based image formats (GIF, WebP, PNG, etc). Follow my progress here: https://twitter.com/mattdesl/status/1346048282494177280?s=21