My suggestion would be to DIY. That's what I do here for most of my data. I do it straight on top of Linux + ZFS (put your right hand up and your left hand over your heart and repeat with me, three times with gusto: "ZFS is absolutely great!"), but if that's too geeky for you, there's ready-made solutions on top of them (or similar) that encapsulate most of the complexity for you, like FreeNAS or Proxmox.
Hardware-wise, there's a lot of choices to be made, but in my case (as I don't need all the data online all the time) I partition it on sets of 20TB each, and store it in sets of 6x 5TB "portable" USB3 external HDDs (like this one) formatted as ZFS raidz2 pools. These disks are then connected to my main computer and used as needed. To be able to find which set hosts a specific file or directory, I have a "master" text file on my computer that's basically an aggregate of the list of files/directories on each site (ie, find /POOL_MOUNTPOINT -ls >>~/master_file.txt).
Each 5TB HDD costs about $100 new (and cheaper when you can find good used ones in eBay), so about $600 for each 20TB. If that's too rich for your blood, you can use larger disks, which are usually cheaper per TB, like this 14TB Seagate "Desktop" that can now be had for about $200, so 6 of these in raidz2 would give you 4x14= 56TB of net space for about $1200.
If you need your 200TB online all the time, then things start to get more complicated and, depending on how you want to balance simple/cheap/good/fast (just remember you can't have all four) there's a ton of other options from using cheap hardware like Raspberry or NUC small computers plus USB hubs and external drives or multi-drive USB enclosures, to outdated-but-still-good-enough used server-class hardware (which can be purchased 2nd hand relatively cheap on ebay etc) plus extenders and secondary cabinets.
Any of these options would IMO/IME give you a much cheaper and more flexible and open solution than a brand-name NAS like Sinology/QNap etc. But then some learning will be required, and the whole she-bang is bound to be more complex to admin (yep, admin -- these things can almost run themselves, but not totally) and problem-prone (depending on the quality of the hardware you end up using), so pick your poison.
Hope the above helps!
EDIT: see this other topic for the guy who assembled (and then colocated, but it would also work for non-colocation) his own 'semi-pro' NAS: Cheaper than cloud storage - colocating my own storage server