That's pretty hard to do, since nearly everything in IT today is web-related: enterprise client/server applications; mobile apps; the burgeoning Internet of Things (IoT).
...with full awareness of the irony involved, I will note that I actually did have to write some JavaScript code last year, for work - not because I was doing anything related to the web, but because I was building some test cases for our static-analysis tool. So even in this nearly web-free paradise, JavaScript still managed to wriggle in.
There's no doubt this policy has created some interesting challenges for my career path, but I still think it was the right decision. It reduced the number while increasing the quality of the available work options.