I am not a Doctor, but my wife works in the OB/GYN field. I get called in for "emergency issues" in my field and my wife gets called in for emergencies as well. In my opinion the stress level will depend your skills, length of time in the field and personality. For instance the first time I was called in for a Major IT outage I was scarred because I did not know what I was getting into and I was unsure of my skills. 20+ years later, when I get called in for an emergency I am confident that I can resolve just about any issue and now it is more about lack of sleep... Now there is a difference between an IT Emergency and an emergency delivery of a human life, however several Dr. friends tell me it is similar for this reason. After 20 years of doing anything it becomes second nature. Think of driving a car. The first time is terrifying, After the first week or two you feel comfortable. After 6 months you are confident, until someone slams on the brakes in front of you... That is your emergency... That will bring you back to your "first time is terrifying" feeling. After years and years of driving when someone slams on the brakes, you tap on the horn and move on.
All of that being said, there is a debate who said this quote however, Choose a job you love, and you will never have to work a day in your life.. If OB/GYN is what you love to do, Do it. The stress will melt away after some experience. Good Luck.