TL;DR: An Inwin IW-MS04, Fractal Design Node 304, or Silverstone DS380.
For the purpose of this, I'll be pretty much sticking to 3.5" HDDs–both since that's in the question, and since 3.5" HDDs still contain several advantages. Though high density 2.5" drives are becoming more and more common–@Yisroel Tech pointed out a 5TB drive–they're still not what I'd consider "ready." As of yet, most of these drives are intended as secondary for laptops: high density, 5400RPM, desktop drives (read: not rated for 24/7 usage nor-more importantly–high vibration environments.) They're also generally much more expensive than their 3.5" counterparts, especially given the ability to shuck WD Reds from external enclosures. Though 2.5" drives have also become common in the data center, these are high speed SAS drives, not SATA, and don't really make sense for a desktop build, especially when price is at a premium. In general, they're still also more expensive than their 3.5" counterparts.
Basically, I went through PCPartPicker's list of Mini ITX cases, then for each case that looked promising, looked up the manufacturer specs on volume. For sake of transparency and record-keeping, here's the table I produced (you'll need something to render Markdown tables with, since StackExchange still doesn't support them.)
Case | Bays | Volume
--- | --- | ---
*[Lian-Li PC-Q26](https://pcpartpicker.com/product/fnQRsY/lian-li-case-pcq26a)* | 10 | 34L
**[Silverstone DS380](https://pcpartpicker.com/product/BrtWGX/silverstone-ds380-mini-itx-tower-case-ds380)** | 8 | 21.6L
*[Lian-Li PC-Q25](https://pcpartpicker.com/product/zth9TW/lian-li-case-pcq25a)* | 7 | 20L
[BitFenix Phenom](https://pcpartpicker.com/product/37mLrH/bitfenix-case-bfcphe300wwxkkrp) | 6 | 31L
*[Fractal Design Array R2](https://pcpartpicker.com/product/9KrG3C/fractal-design-case-arrayr2)* | 6 | 17.5L
**[Fractal Design Node 304](https://pcpartpicker.com/product/BWFPxr/fractal-design-case-fdcanode304bl)** | 6 | 19.5L
*[Lian-Li PC-Q08](https://pcpartpicker.com/product/2w2kcf/lian-li-case-pcq08a)* | 6 | 21.3L
[BitFenix Prodigy](https://pcpartpicker.com/product/KsTmP6/bitfenix-case-bfcpro300kkxskrp) | 5 | 26.3L
[Lian-Li PC-Q35](https://pcpartpicker.com/product/zv648d/lian-li-case-pcq35a) | 5 | 18L
**[Inwin IW-MS04](https://pcpartpicker.com/product/7f7CmG/inwin-iw-ms04-mini-itx-desktop-case-iw-ms04-01-s265)** | 4 | 13.3L
[CHENBRO SR30169](https://pcpartpicker.com/product/ncqbt6/chenbro-case-sr30169) | 4 | 16.7L
*[Rosewill Legacy V6-B](https://pcpartpicker.com/product/KJYXsY/rosewill-case-legacyv6b)* | 4 | 12.6L
[Antec ISK600](https://pcpartpicker.com/product/rVh9TW/antec-case-isk600) | 3 | 18.7L
[Cooler Master Elite 110](https://pcpartpicker.com/product/QCjG3C/cooler-master-case-rc110kkn2) | 3 | 15L
[Cooler Master Elite 120 Advanced](https://pcpartpicker.com/product/VrZQzy/cooler-master-case-rc120awwn1) | 3 | 20L
[Cooler Master Elite 130](https://pcpartpicker.com/product/6wR48d/cooler-master-case-rc130kkn1) | 3 | 20L
[Cooler Master HAF Stacker 915R](https://pcpartpicker.com/product/Xg2kcf/cooler-master-case-haf915rkkn1) | 3 | 32.5L
[Cooler Master HAF Stacker 915F](https://pcpartpicker.com/product/hRQypg/cooler-master-case-haf915fkkn1) | 3 | 32.5L
[Thermaltake Core X1](https://pcpartpicker.com/product/8238TW/thermaltake-case-ca1d600s1wn00) | 3 | 56L
[Fractal Design Core 500](https://pcpartpicker.com/product/LvnG3C/fractal-design-case-fdcacore500bk) | 3 | 19.5L
[Raidmax Element](https://pcpartpicker.com/product/HJVBD3/raidmax-case-atx101bup) | 2 | 16L
<s>[Antec ISK 300-65](https://pcpartpicker.com/product/2YGkcf/antec-case-isk30065)</s> 2.5"| 2 | 7L
*[Inwin BP671](https://pcpartpicker.com/product/fmyFf7/inwin-case-bp671fh200b)* | 2 | 8.2L
[Thermaltake Core V1](https://pcpartpicker.com/product/fGvRsY/thermaltake-case-ca1b800s1wn00) | 2 | 45.5L
[Corsair 250D](https://pcpartpicker.com/product/gHkD4D/corsair-case-cc9011047ww) | 2 | 28L
[Lian-Li PC-Q19](https://pcpartpicker.com/product/bp98TW/lian-li-case-pcq19b) | 1 | 22.5L
*[Silverstone RVZ01B](https://pcpartpicker.com/product/GCfp99/silverstone-case-rvz01b)* | 1 | 14L
*[Silverstone FTZ01](https://pcpartpicker.com/product/P9Wrxr/silverstone-case-ftz01b)* | 1 | 14L
[Skyreach 4 Mini](https://www.sfflab.com/products/nfc_s4m) | 2x2.5" | 5L
[DR ZABER Sentry](http://zaber.com.pl/sentry/) | 4x2.5" | 7L
[Louqe GHOST S1 MkII](http://www.louqe.com) | 3x2.5" | 8.2L
[DAN Case A4 SFX](https://www.dan-cases.com/dana4_spec.html) | 3x2.5" | 7.2L
From that, I chose the above four cases, primarily based on drives/volume and availability (since some of the cases have unfortunately been discontinued.)
At 13.3 liters with support for 4x3.5" HDDs hidden behind a pretty face, this case is pretty impressive in itself. What's more, these drives are hot swappable. However, this does come at a cost: though it comes with a 265W Bronze PSU, you probably won't be able to upgrade from this if you decide to, since the case only supports Flex ATX power supplies. The $160 price tag doesn't help either. Oh well. Still worth a look if you meet the power requirements!
At 19.6 liters with support for 6x3.5" HDDs with support for full ATX PSUs up to 160mm in length, this case sure packs a lot into a little package, and for $100, isn't too expensive either. Fractal Design also has an awesome reputation and good looks.
Unfortunately @Nicholas Martin got to suggest this before I could, so I'll just have to second the quality of this case. At 21.6 liters with support for 8x3.5" HDDs in hot swap bays, this is the best in drives/volume. It's not too expensive at $150, but you will need a more expensive SFX or SFX-L PSU. Silverstone also has a solid reputation and good looks.
Sadly, there are several cases which I would have loved to recommend, but are discontinued and out of stock: the Lian-Li PC-Q26, Fractal Design Array R2, Lian-Li PC-Q08, and Rosewill Legacy V6-B. If you can find one of these used, they're amazing options, but my guess is they were discontinued due to low volume sales, as they're fairly specialized products.
The Inwin BP671 is a cool little case at a mere 8.2L that supports 2x3.5" drives. However, the bundled 200W TFX PSU is just too weak and too sketchy for my recommendation, and availability of TFX PSUs is quite low.
The Silverstone RVZ01B and Silverstone FTZ01 are both neat little cases at only 14L, but 1x3.5" drive is just too little for NAS purposes.
Really small cases
If you were paying extra close attention, you'll notice at the bottom of my table I listed three cases that don't even support 3.5" drives! Okay, yes, I said I wasn't going to do this, but I had to mention them if not recommend them. They're all extremely tiny, extremely well built, extremely expensive cases. Even though they're not really appropriate for this application, they're interesting to look at.