How do you move a 650lbs Rack on Hardwood...

I think the guys who suggested the Masonite is the way to go. Don't forget the total weight is divided by the number of casters. Assuming you have 4 casters, you'll only have about 165 pounds on each. No problem for 1/8 masonite.

I would use 2ea Hand Carts with phneumatic tires. Place lip of each cart at each end of rack. Place thin piece of something on bottom of rack to prevent the cart's lifting lip from scratching anything. Use the lever action of the carts to lift high enough and roll it on over to where you way it. Have a 3rd person steady the rack.

If the floor is smooth and level, just a steel plate on the bottom of the rack with a single 3/8" hole in the middle attached to compressed air source. Assuming the footprint of the rack is 21" x 24" = 504 sq.in. Divide 650 lbs by 504 sq.in. = 1.3 lbs/sq.in. not that big a deal really.

Get those plastic tracks that refrigerators are moved on. You lay them on the floor and the fridge rolls/slides on them. Like the furniture gliders, only on about 2.5' tracks that are about 4" wide. They interlock too if you need to go further distance. They'd likely be at any home improvement store. If the rack ends up in the closet ontop of them, so what? Ready for removal at some later time.

After reading the various options, I vote for the masonite as the cheapest and safest solution. K.I.S.S.