Do people really only keep 3-6 months of living expenses in their savings?

In most articles I've seen, it is recommended to keep 3-6 months of living expenses in a savings account, and invest the rest. This seems so.... dangerous to me. Do people really do this? That would be less than 20k in savings. What do you do if you suddenly need a new septic system, a new roof, etc? Am I catastrophizing?