Long story short, I made a silly mistake.
I had setup my QNAP NAS to use the IP address 192.168.1.1/24 and 192.168.1.254 as the default gateway.
Using the web configuration interface, I then changed from this network to another and the first thing I did was to change the static IP to 192.168.123.1/24. Unfortunately, the default gateway setting isn't available in the same place, so I just thought I'd change it just after setting the new IP and netmask.
This locked me out of the NAS: when I try to reach it using HTTP on its new IP adress, the packets are routed correctly, but the NAS never responds (I think that's because it's instructed to route via 192.168.1.254, which is on a different subnet than its network interface, so the packets never leave the NAS).
Is there any way I could reach it again via HTTP ? I tried different things using Wireshark to debug and multiple interface settings on my computer but I never receive responses to the TCP packets sent to the NAS.
Interesting fact: ping works. It don't know ICMP very well so I don't understand how the responses are sent to my computer, but they sure are!
As a sidenote, I can't reset the network configuration of the NAS as I had disabled the quick reset via the reset button. The ultimate step would be to extract the disks, copy the data and start a new configuration from scratch.