Liability insurance isn't strictly unlimited, the coverage limits are just very high - e.g. in Germany, personal damages are covered with a minimum of 7,5 mio. euro (with a maximum limit of 8 to 15 million per person, depending on the insurance company), and total damage is mostly limited at 50-100 mio. euro. For most accidents, these numbers are fairly equal to "unlimited".
About health insurance - I'm not quite sure how you come at the conclusion that state pays everything. Injuries in an accident are paid for by the liable party's liability insurance, the same as in the U.S.
In other countries i paid that per year for way more peace of mind.
But yesterday i did the car coverage for a used 09 bmw 128iC.
Couldn't get any lower than that on several websites
I have two vehicles less than 5 years old, with two drivers, with way more than $300k coverage for liability and comprehensive, and a $0 deductible by one of the top/largest insurance carriers in the U.S. and I pay $75/mo, total.