I do not believe you did something silly. The operating system, here also called "firmware" is not placed on the disk, it's in the internal flash memory of the device. So, I can guarantee you that my devices boot without a disk (even if this is not useful for storing data
Typically the formatting takes some time and after that, the device can be used again with the disk empty. I am not sure at the moment if the formatting of the disk via the web interfaces causes an automatic reboot or not, and I do not want to test this with my Storagebird LAN 2 now because I have data on it...
Maybe your Storagebird LAN 2 rebooted or crashed for some reason and forgot it's configuration. Then, it maybe has got an IP address via DHCP or it has got the default IP address 169.254.0.1 (according to the manual). You should check if the device can be addressed with one of these possibilities. If you do not have any warantee on the device, you can try to open it and remove the hard disk. If the device cannot boot because of a hard disk failure that occurred while formatting, then this might help locating the problem.
By the way, the forum "Storage solutions" normally does not treat the home NAS devices, so to get better or sooner responses, you may want to try the Accessoires part of this forum or you can also visit the digital home forum (see my signature), there is a lot on Storagebird LAN 2.