Remove NoData from ArcGIS Raster Dataset

Long story short you can't. Raster data is a continuous grid, a perfect square or rectangle. You can't poke holes in it.

The No Data is designed as a placeholder for areas where you have no data such as a mask or areas where there is actually no data. Most software will transparent it out for you by default. But as far as completely eliminating it from your dataset I don't think thats possible.