This is one of the factors driving the development of orbital satellite data centers. While satellites are also vulnerable, they are at least somewhat harder to hit than a stationary terrestrial data center. For now it's questionable whether such constellations are feasible due to concerns over launch costs and cooling but some people are taking it seriously.
Batch processing could be done in solar orbit.
The difference is that to hit stationary terrestrial data center in the USA you'd need to go through multiple layers of defense systems, not to mention it would probably be an act of war.
As for the satellites they can be jammed and there's not much you can do about it.
In addition to all the engineering problems associated with cooling those data centers and the shields that you'd need against cosmic rays, those things would only accelerate Kessler syndrome.