I think it's an anxiety problem which you can over come thru therapy. You need and want to be able to help your family when they are sick, and this goes beyond normal fear, usually nothing will stop us from helping our loved ones. See a professional to learn coping skills and to understand why you have this fear. When fear paralyzes you to this point, there is more going on. Good luck to you.
Too much empathy. My dad used to throw up whenever one of the kids did. When my brother broke his collarbone, I was so upset I had to sleep in another room, and it wasn't even my bone.