Why, Lord, is it so difficult?
Why did you reboot?
If - when you disconnected the DNS-323 from the network the disk activity stopped, the probability is that it is caused by access from an external source, something on your LAN is accessing the unit - this would have been confirmed if the activity retarted when the network was reconnected.
Yes - I recognize that you said all the computers on the network are off - have you considered the possibility of an intruder - someone connecting via an insecured wireless LAN, perhaps you have the ftp server turned on and a port forwarded through your firewall.
The idea here is to isolate the problem it's either internal to the device or external - if it continues with the device disconnected from the network, it's internal - if it stops when the device is disconnected, it's probably external, and then you are left with the task of determining what causes it - use a network sniffer to examine the traffic to the device for clues.