I have a full-height server rack with large, loud, noisy, power-inefficient servers, so I can't provide much of a good suggestion there. I did want to say that you might want to seriously reconsider using a single 10Tb hard drive.

Hard drives fail, and with a single drive, in the event of a failure, your data is gone. Using several smaller drives in an array provides redundancy, so that in the event of a drive failure, parity information exists on the other drives. As long as you replace the failed drive before anything else fails, you don't lose any data. There are multiple different ways to do this, but I'll use RAID as an example. In RAID5, one drive stores parity information. If any one drive fails, the array will continue running (albeit slower); you just need to replace the failed drive and allow your controller to rebuild the array. In a RAID5 configuration, you lose the space of one drive to parity. So if you have 4 4TB drives in a RAID5 configuration, you would have a total of 12TB of usable space. RAID6 lets you lose two drives, but you also lose two drives worth of space to parity, meaning your array would be more fault-tolerant, but you'd only have 8TB of space.

There are many different RAID configurations; far too many for me to go into them all here. You also have something called ZFS, which is a file system with many similarities to RAID (and a LOT of extra features... like snapshots). As an example, I have 12 10TB hard drives in my NAS. Two groups of 6 drives are configued as RAIDZ2 (similar to RAID6), for a total of 40TB usable space in each array. Those two arrays are then striped (like RAID0, so that data is written across both arrays with no redundancy at the striped level). In total, that means I have 80TB of usable space, and in a worst-case scenario, I could have 4 drives (two on each array) fail without losing data.

I'm not suggesting you need a setup like mine, but you could probably fit 3 4TB drives in a small case, use RAID5 or ZFS-RAIDZ1, and still have some redundancy. To be clear, RAID is not a substitution for a backup, but it can go a long way toward ensuring you don't need to use your backup.

For home lab/hosting you can get really good performance out of used/refurbished mini pcs, look for something like hp mini 400 g4. They can be found for around $100 to $150 on ebay. They generally fit a 2.5 inch ssd drive as well as 1 or sometimes two m.2 ssd drives. I find that intel nucs are pretty expensive compared to the specs, like you're paying a premium for the brand and form factor since it's a bit of a niche market. These mini pc's are more business office oriented and are way more accessible at a lower price.

I realize you're asking about storage, you could go the route of turning that old server into just a dedicated nas and use other hardware for your services/compute.

TBH, you need additional backups anyway and you don't need 100% uptime. You don't need to pay much for it for internal redundancy (aside of storage) and server features.

Buy the nuc, buy one or two 10GB HDD with usb/usb-c and an external case and your are good to go.

But in the end, any PC will work. Get a cheapo PC, buy 1 SSD for the os and container and a big external disk for storage.

I'm running a Ryzen 5 5600G with 32GB Ram, B450 Chipsets, 1TB SSD for internal and 2 4tb SSD for storage in a micro atx. Cost me around 1200€ bucks 2 years ago.

You can even use an old laptop with a broken panel if you like.

Another possibility is, to buy a cheap (old) server from some company renewing it's support contracts (loud, space, power hungry).

Or buy an fanless industrial pc. Anything is possible. You could even try to use arm.

I'd recommend to optimize for power consumption, noise (depending where it is located in your home) and storage. Not ECC or redundancy. as long as you do regular backups to another system, which you should do in any case, there is no reason to pay double for something.

First thing I bought after my system was running, was an additional nic, to be able to use the server as a firewall, not an UPS or another hot standby PSU

If your requirement is a 3.5" drive bay, then maybe check out some of the SFF form factors, like an HP Elitedesk SFF 800 (not the mini/micro). It has the same hardware as the mini, takes the same amount of power, but has a space for a 3.5" disk and a slimline optical disc. It's bigger than a NUC, but smaller than a tower.

You are correct, they are limited to 2.5" drives. They also take m.2 and if i'm not mistaken those go up to 8Tb and 5Tb for 2.5 SSD. Not the ideal single drive solution but if you had a total size goal it looks like you could hit it with an NUC