By far the best way to store larger ropes, like climbing ropes, is to simply stuff them into a bag of the appropriate size, securing the end at the mouth. The rope will feed flawlessly from the bag, whether thrown or simply pulled out gradually. I have used this method for ropes as long as 600 feet, and I understand this is the technique used by Yosemite Search and Rescue to store and deploy their 3000'+ ropes, which span El Capitan top to bottom.

I have also used this technique on paracord and it works quite well. It also works with kayak throw bags.