It is going to involve tradeoffs. Number one is that you are probably not going to wind up with a fanless NAS. 5 drives that each dissipate maybe 10 watts means 50-60 watt TDP for your proposed system. You cannot cram something generating that much heat into a NODE 304 without forced air cooling, or it'll just bake.
Most Atom boards I've seen use crummy ethernets like Realtek. The Realteks typically require a fair bit of CPU to operate, compared to Intel parts. Usually Supermicro is the main manufacturer offering Atom-with-Intel-ethernet. They do have the X9SBAA. Which looks like a sweet little board, but:
1) Only 4x SATA3 ports
2) Only one expansion slot, and that's PCI 32-bit, you'd need to put your fifth drive on an add-on PCI SATA controller.
The Atom does not have a ton of oomph and will never make for the fastest of NAS options. On the PassMark test it
can't even break 1000, only managing 916. It will be similar in speed to an N36L or N40L, which there are plenty of discussion here on the forums regarding performance for various use cases. Please feel free to search for more detail.
Considering that the board in question seems to retail for $220-$250, for a PassMark of 916, I feel compelled to point out that a Supermicro X9SCM ($160) plus an Intel G2120T (PassMark 2850, 35W TDP, $60-$75) winds up being the same price. It'll give you a lot of options for future expansion that the Atom will not. The Sandy and Ivy parts seem to have good energy management. With a 35W TDP you can make arrangements to avoid a CPU fan, but as previously noted you aren't going to get away without some sort of system cooling.
