Ah the joy of external RAID in a box...
Be forewarned: I am not trying to insult you or your choices. I do, however, hate these kinds of devices and I'm not going to make any apologies for them - they're generally overpriced cheap junk, and often fail more consistently than whatever they're advertised as being a good back-up device for.
So, it sounds like the "thing" here is not doing so well. Opening it needs to be the next step - you can evaluate the drive's health with them removed. Most likely the little controller board in this thing (which is very probably made from the absolute cheapest possible components, usually installed in the cheapest possible way, with no consideration given to cooling or longevity, it just has to make it out of the factory) has seen better days, and that's why you're having problems (I honestly can't tell you how many of these things I've junked over the years for this exact reason - they are *not* reliable devices and they do not stand up well to running 24x7). But wait it gets better! The drives are in RAID, but this may have been (as in: very probably was) accomplished in some proprietary way (as in you can't get that controller elsewhere and the data is basically chained to it), so recovering data stored in there may range between impossible and total pain in the neck (and yes, I hate these devices as a general rule, for this specific reason - they take in a ton of data and then leave you completely hosed when they fail, and often the manufacturers are all too happy to offer you some hilariously expensive "recovery service" that usually gets you a new piece of working hardware and that's about it; you could add the whole "lets throw multiple terabytes over USB" as a close second).
My advice at this point would be to open'er up and see what you can see. Sniff around for obvious signs of burn-off. Clean out any hairballs, dust colonies, etc as you find them. See if it runs better with its chassis open (no you won't let out the magic smoke, but some of these Buffalo enclosures are put together in such a cheap manner that the primary enclosure cabinet also encloses the PSU, so don't go sticking your fingers where they don't belong). If the drives have been over-temp'd and/or the controller board has been cooked a bit, it isn't actually that uncommon to experience what you're describing. Yes it will take ages to get the data off of there, but that's honestly "step 2" - get the data onto something safe and secure and then you can toy around with more intensive testing on the drives themselves (so that you aren't risking your data). The reason I say open it up and clean/try to get it better cooled is that it may help the file transfer situation.
So where are you putting the data? Well, an actual server is always a good choice. But we live in the era of disposable mobile devices and everyone wants their cute little USB gadgets. So the easy second answer is to grab a big'ol single-disk external solution, sure it isn't going to be a bastion of reliability or data security, but there's also no proprietary tomfoolery behind it, and the data is much more recoverable and transportable. Modern drives exist that can easily hold all of this data on a single disk - you're talking about 3/4ths of 8TB if I'm not mistaken, but even at a full 8TB you can do that to a single drive without fuss.
And as far as testing the drives? You'll need an external enclosure if you don't have a desktop with the correct ports. You'll also need to know who made them. And then use the appropriate diagnostic tools from that manufacturer (be lazy, use Ultimate Boot CD, but you still need to know which tool you're booting into). It may be that the drives themselves can be salvaged, and with luck its just the chassis that's toast, but the drives may also be a lost cause. Just preparing you for the worst case scenario (and again this is why I'd get the data onto something else for its own safety (and on that note: RAID is NEVER a substitute for a proper backup regime - multiple copies make for a happy life)).
h34r:
