There's a raspi nas card so you can make a raspi a NAS with a lot of sata ports, like 8 or something! It's called pinas or something. I found a tutorial for creating your own raspi nas but it's not with the sata card I saw before... Here's a link :) When you search for "convert raspberry pi into nas" you will find a lot of tutorials and tips.

I'm at a Christmas party with friends so I'm not going to search dive to find the thing I saw right now, but know there's a sata expansion card for a raspi, I hope you can find what you are looking for!

Making a hole in your closet is something I can recommend, I'm autistic, I love cable management. But if it's an antiques cabinet I wouldn't do it. Also, if it's Ikea or something similar (laminated pressed wood fibers), when you make a hole you break the seal in the top layer so moisture can get in. This can lead to mold in the pressed wood fibers. When your house isn't cold during the winter and moisture free, it's not a big issue. Just add small planks on top and the bottom with clamps when making the hole and drill through that, otherwise it will splinter and get ugly. If it's solid wood, I would definitely make a hole to make the cables invisible.