1. Most hard drive noise is caused by mechanical vibrations being transmitted to the chassis the drive is mounted on.
2. Consequently the most effective way to reduce noise is to reduce the mechanical coupling in the drive mounting mechanism. Having the drives in a noise-isolating case is helpful too, but only as a secondary improvement. Optimizing the drive mounting should really be the first priority.
3. If space isn't a concern the optimal thing is to have a large case (like an ATX or larger) with a large number of HDD bays. The mounting should use soft rubber or silicon grommets. Some mounting systems can work with just the grommets, but systems that use screws are ok too as long as the screw couples to the grommet not the chassis. In a good case like this any number of hard drives can be made essentially inaudible.
4. If space is a concern, a special purpose "NAS like" case (example: the Jonsbo N line of cases) can approach the size of consumer NAS boxes. The lack of space makes optimal accoustics difficult, but it will still be a 10x improvement over typical consumers NASes.
5. Lastly what you shouldn't ever do is get one of those consumers NAS boxes. They are made with no concern for noise at all, and manufacturing cheapness constraints tend to make them literally pessimal at it. I had a QNAP I got rid of that couldn't have been more effective at amplifying drive noise if it had been designed for that on purpose.