That depends on the tires you are driving on. there needs to be at least an inch of ice on the roadway before you really need chains. Today the All-Season tire can handle a little ice. Just remember the rules of driving on ice.
Only go as fast as you are comfortable. Keep to the right if possible and let the bozo driving 50 around you. If you hold up traffic a little don't worry about it.
Chains may not be allowed on the road as they definatly tear up the road. If driving from San Francisco to Tahoe for example. You will be pulled over and told to remove the chains before you reach the Golden Gate. Toss them in the trunk and put them on IF needed. Practice where it is warmer so if you have to it is a quick and simple task. They usually take 15 minutes/tire to apply. there are traction straps you can get also for fron wheel drive and just clip on real fast.
If you have a front wheel drive you really don't need chains as the car is pulled rather than pushed and the chains would go on the front wheels and that can interfere with the fender and cause serious body damage.
The way I read it, 600 miles with chains, Don't do it, it will tear up the cord in the radial tires. If you are serious about extra traction on that long a trip get snow tires. Most of the time in winter months they are allowed on highways and the ride while noisy you will have the extra traction you need in the ice.

Chains can do damage if they are not fitted properly. Not only to the tires but also to the under-carriage of the vehicle. I've heard of loosely fitted chains eventually breaking away and getting wrapped around axles.

However, if installed properly, they should cause no harm to the vehicle.