Okay, full contact karate and BJJ can have their moments. But if you're interested in martial arts without all the GRRRR, there are options!
Allow me to introduce: AIKIDO
Aikido is known as "the gentle art." The class has more of a Yoga feel than a competitive one. Aikido focuses on joint manipulation and off balancing. In training, you learn how to feel the momentum of your attacker and then use it against him/her. Students practice on training partners, but in a non-competitive way.
Instead of competitions or tournaments, Aikido has seminars. Sometimes guest instructors will come in and give lessons. Other times the class will travel to another school. Joint training sessions with other area schools are also common.
When people are nervous about taking a martial art and fear injury, I usually guide them to Aikido.
That being said, you often get out of training what you put in. BJJ and Full Contact Karate (Ashihara) can be done gently and without injury. Just like it is possible to get injure someone if you land an Aikido technique full force. But if you're looking for a martial art that is gentle, non-competitive, but still can be used for self defense and personal development, Aikido might be a perfect fit. We have Aikido for Kids and Adults.