Dealing with a chmod gone wrong

I could use some advice to unfuck a system. My colleague shot himself in the foot with Linux and now at least his Docker doesn't work properly anymore, but god knows what other issues will come up later.

In a dev project he executed sudo chmod -R 755 /var where he used /var instead of the local var folder. He doesn't really understand Linux, and now it's my job to fix it :-(

Now when I start docker containers it seems successful, but docker ps doesn't list them and I caot enter them with docker exec because it says that they're not ruing.

The docker problems are probably only the first symptoms. Does anyone have any advice on what I could do? How bad is this?

System is Ubuntu 14.04.

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