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The answer is going to be expensive no matter which way you go.
Even a self-build will be expensive as there are few cases that can take that many drives, and server cases (racks) are very loud and power hungry.
But for a NAS, look at the Synology DS 1821+ https://www.synology.com/en-uk/products/DS1821+ which supports 8 HDD and can be extended with a couple of expansion units https://www.synology.com/en-uk/products/DX517 to give you 28 HDD.
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Cases that take 8+ 3.5" drives, cheapest first:
https://uk.pcpartpicker.com/products/case/#J=8,20&sort=price
You can also buy extra drive cages for a lot of cases, guess that is not reflected in these listings.
I think what you are looking for is increasingly obscure and priced accordingly, though.
I'm finally getting round to updating my NAS. Looking for a case that can take at least 8 drives (mainly 3.5") and preferably more.
Fractal Design have a few but does anyone have any other suggestions? Googling it mainly brings up loads of US suggestions that are unavailable or really expensive over here.