However, if the servers use external storage sources, the data may be corrupted even if the equipment continues to function. Also, for servers that are network booted, if the network components are damaged, it might be impossible to reboot the servers even if they themselves are undamaged.
You are more of a tech fan than I, so I take your word.

I do suspect that most critical off site storage is caged also - it costs virtually nothing extra to cage a vault. Physical media, such as DVD's will be unaffected. Much of the old ARPAnet is not only still functional and in use every day, but hardened. Fiber optics will be unaffected. The transatlantic cable has a couple of miles of water shielding it. and water is a very good damper for ions.
The capacity will be there, but the question is will anyone be able to use it? Might be an idea to grab one of your old laptops and a cheap metal cabinet. Put the laptop in the cabinet and ground the cabinet to any convenient pipework.