Insuring a vehicle with a salvage, rebuilt or flood title is not impossible. The main thing to do when considering the purchase of a vehicle with a salvage, rebuilt or flood title, is to call multiple insurance companies to find out if they do in fact insure vehicles with these sorts of titles. Some companies may only insure one type of title but not the others.
My own experience had to do with a 2010 Lexus ES 350 flood damaged total loss vehicle. The seller had sold many total loss flood damaged vehicles (bought at dealer auctions) and explained to me why the car was not such a risky purchase. The vehicle's appearance was great. It had 65k miles on it and the seller was asking $7k below market value for it. Sounds great right? He explained that the flood water had not reached the electronics at the dashboard level where the most damage would occur in a flood situation. The flood water had reached a couple of inches above the door threshold, nowhere near the main electronics. He had removed the carpet, checked for rust, replaced the carpet and had driven it around for 500 miles. My only concern about buying this car had to do with the power seat electric motors which the water had definitely reached. Replacing those for the mechanically inclined, or with the help of a YouTube video, is an afternoon DIY job. I was ready to buy. All I needed was to find a company to insure it.
I called 3 or 4 companies and found one that would insure it as long as I verbally stated that the car had been fully repaired. They did not care if it was a salvage, rebuilt or flood loss vehicle. Great, right? Well, the catch, as you may have thought, was the cost. I wanted to insure at full market value of 16k with collision. 9k was a bit much for me to risk on liability only. This would be car number three for our family and I was on a budget, so I decided to stick with 2 cars and adding my 16 yr old to our policy. The seller told me most of the flood damaged vehicles he had sold that year (80 in all) were Hondas or Toyotas at price in the $3k - $5k range. I think this is definitely worth the risk of liability only coverage.
Unless you purchase a damaged vehicle from auction yourself and can oversee the repair process yourself, I would stay away from rebuilt and salvage titled vehicles. Buying a flood title vehicle? Well, that might just be worth the risk.