We are not able to tell you if your specific herniated disc is safe with any particular exercise, since we are not privy to your exact condition and medical files.
It is important to talk with your doctor, and better yet, meet with a physical therapist who can SHOW you the appropriate exercises for you! Not only do you want to do the appropriate exercises so you no longer "feel like mush", but there are specific back exercises that can help strengthen your back and relieve symptoms...aiding in healing.
If you are hesitant to do talk with a PT, you may choose an activity such as gentle stretching to help with your range of motion, but again: you need to get the "OK" from your doctor. Some experts have suggested using the shallow end of a pool to help with stretching.