If you owned a car with a salvaged title, can you get insurance? Probably not. Why not?
The insurance companies are required to put the car back into the position it was in prior to the accident. If the car is salvaged, it is almost impossible to put it back into the position it was in prior to the accident. And, any damage totals out the car. So, it would be hard for them to determine a value on the vehicle. However, this is true only for comprehensive and collision.
For liability coverage, it is out there. You need to shop around and you will end up with a non-admitted or non-standard carrier. I know I generally tell people to stay away from insurance companies that are not admitted. But, in this case, similar to insuring a classic car, you need to shop around for a policy that fits you. And, this is going to be a specialty type insurer.
So, call around for liability insurance and do not take no for an answer.