Its a bit tricky. There's three elements to your problem and each of this needs to be handled seperately. Data, Power and 'space'
Easiest way? USB 3.0 Hard drives. They are typically fast enough for bulk storage these days, though clearly you arn't going to get blazing fast speeds. A 2.5 inch drive goes up to 2tb, which is pretty impressive data density, and this is the neatest solution. You could also get an enclosure with more drive bays connected over NUC. Newer NUCs would do thunderbolt wihich is even faster.
This is the sane way to do this. I did once figure out how you could get 3 drives into a NUC tho. This is entirely theoretical and somewhat impractical IMO.
If you need more speed, you can trade your wifi adaptor for more storage - a mini pci-e adaptor can be broken out to sata for 2 more drives, which takes care of your data.
Power is tricky - You might be able to split the internal sata power adaptor or get a power brick specially designed for hard drives. Split this and you have power for the hard drives. They are laptop drives so power needs might be low but something like this should cover it handily
Space? Intel has an option to 3d print lids. Design a new lid to hold the drives, 3d print it. Hard drives are pretty standardized.
Personally, the USB 3.0 HDD option or external drive bays are the way to go, especially since you can turn off unwanted drives as needed.