I've given serious thought to adopting a Greyhound, and have talked to some rescue people, as well as other Grey owners. Greyhounds are quiet, calm dogs--absolutely useless as a watchdog or guard dog. Around the house, they tend to be couch potatoes. I've heard varying statements of how much exercise a retired racer needs; it probably depends on their age and overall health. Your best bet, if you decide to adopt a particular dog, would be to check with the rescuer or foster parent of the dog for exercise requirements. A fenced in yard, with AT LEAST a 6 ft fence is a must. An enclosed dog run with a lid might be better. Again, I've heard conflicting information about this--one person says they don't jump, another says they will. They can never, ever be walked without a leash! Health is generally good--they are prone to certain types of cancer (bone cancer). It is essential to find a vet who is accustomed to working on sighthounds, as they do NOT tolerate anesthesia well, and even a routine teeth cleaning can be fatal from the anesthesia. They may have very poor teeth--dental problems are common to the breed (maybe due to what they are fed at the track). Due to a very thin coat and little body fat, they are susceptible to cold, and a warm jacket is essential on cold days. Their skin is very thin, and they will get cuts and lacerations easily. They may be dog and small animal aggressive.
That's about all I know about it. I am still considering a Greyhound, sometime in the future.