How many drives are you talking about? If my home is burning down, I'd just grab the whole NAS, I mean how big does it need to be unless you have a boatload of drives?
I'm also a bit confused about what you have linked. It's a page to configure a workstation with a starting price of $1749 and goes up from there. That seems on the expensive side for just personal storage that isn't even important enough for an off-site backup (how about mirrored to HDDs you put in a safety deposit box instead of a cloud, or a tape backup and you just grab the tape(s) ?)
Are you getting a raid card or onboard raid with these desktop motherboards and what raid level? Sorry that I'm drifting off topic a bit. 3.5" HDDs or 2.5" SSDs or ?? You can get slide out sleds for 5.25" bays and then any case with a sufficient # of 5.25" bays would work. Many (most) cases have 5.25" bay covers that just pop off if they aren't, or you take out, screws holding the front bezel faceplates on so they are just friction fit, or with cases that don't have a snap off bezel due to the bezel being held on with screws, can take the screws out of the whole bezel, or cut off friction tabs if present, and take the whole bezel off for access, and use whatever is needed for a flat surface interface to hold the bezel on instead of the original screws or friction tabs, for example velcro, or magnets epoxied onto the bezel (if a steel front walled case).
Have you mentioned this to the company you linked, asking what they'd do? You deserve a bit of customer service for the price you'd be (arguably, over-) paying for a mere NAS.
What if you took part of the budget and instead got some USB external HDDs that you keep offline except when making backups to them, then can just grab those to bugout if that day ever comes? I still don't think I'd want to rely on physically being there and able to grab drives to take with me in the event of a site disaster/whatever, but if that was the plan, then the external USB enclosures will at least offer a minimal level of protection more than bare drives would... unless they are SSDs which are inherently more rugged than HDDs.
Further, for your static data (stays the same over long periods of time), you might not need the redundancy of RAID for that data, with the backup copy being on the offline USB HDDs and only raid the drives with frequently changing data. Remember that when you pull drives in a raid array, if they are not just mirrored, you may need the same raid controller to rebuild the array and/or put that data back online to access, while having it on USB external HDDs is about as universal and quick/easy a way to access it as reasonably possible, besides the much slower and less controllable, cloud storage option.