Chronic fatigue syndrome and fibromyalgia are difficult chronic / invisible illnesses to deal with. There are a number of causes and contributors to chronic health issues but the primary two immunity dysfunction and mitochondrial dysfunction. Due to the immune dysfunction and mitochondrial dysfunction this can lead to a lot of fatigue, as well as other symptoms. This makes it difficult for someone that has chronic health conditions to find the energy to work out. Added to that certain exercises and excessive cardiovascular activities could worsen the adrenal system (which may or may not be malnourished already).
I would advise those with any chronic health conditions to consult with their doctor to see if they are healthy enough for exercise. Luckily someone can benefit from even a small amount of daily exercise. A simple exercise one can do daily is walking / hiking anywhere from 20 minutes and up to an hour. It does not even needarily have to be strenuous or fast but can be done at your own pace. If you are often sedentary and work at a sedentary or office its advised to walk everyday to counteract the effects of a sedentary lifestyle. If the job you have is fast paced, with lots of running and walking you only need to walk on your days off.
Strength training is another way someone with chronic health conditions can greatly enhance their health as well. If you follow a high intensity training protocol (HIT), it only needs to be done once a week. Also if done correctly, high intensity strength training will not put stress on your joints and cause a significant amount of pain to your body. A book worth checking out is "Body By Science" by Doug Mcguff and John Little. In it they describe a simple weight training protocol / plan called "The Big Five" which are five simple exercises that target all the major muscle groups of your body. You can then add in additional 1-2 isolation exercises depending on how your body reacts to each workout.
Aside from doing walking and strength training, someone with chronic fatigue syndrome could participate in a hobby or sport if one is well enough. Personally I do dancing and do martial arts. I would not recommend doing a hobby or sport that is too strenuous or difficult for someone that on a daily basis deals with a large amount of fatigue. Other beneficial exercise that are reliably gentle are the body are Tai Chi and yoga, some Tai Chi exercises can even be done sitting in a chair.