Family medicine, depending on coverage of OB patients may not be that stable. Babies get born at all hours of the night, and if they're your patients, you have to go deliver those babies, or have a call system in place so that someone is delivering those babies. Other specialties that don't have a lot of unusual hours include urology, physical medicine/rehabilitation, many internal medicine/pediatric sub-specialties like endocrinology, nephrology, or infectious disease. Other unique specialties within pediatrics like developmental pediatrics, genetics or adolescent medicine have very few, if any, emergencies.
However, a lot depends on how you define "stable" work hours. For example, I'm a pediatric intensivist (Peds critical care). Most people in my specialty are in academic medicine (so they work for medical schools and are involved in teaching/research in addition to their clinical responsibilities). It is a specialty without any outpatient clinic time, caring only for patients in the hospital. For a typical position, an attending physician works 14-17 weeks a year with 36-42 overnight calls a year. If you are at large centers, the call shifts are done in the hospital. The remaining time is spent teaching or doing research.
To me, as someone in this field, this is an incredibly stable set of hours. I even really like the in-hospital calls. I know exactly when I'm going to be in the hospital, and I know well in advance when my call shifts are. There is never a time when I'd have made plans only to have be called in for the night with a sick kid. If I were lucky to get to sleep, then "yay!" I get to sleep, but if not, that's why I'm there and I can sleep the next day. There are "emergencies" but I'm already there, so it's not like it's any sort of burden on me. Those weeks I'm on service are certainly long - being in by 7am, often not leaving until 630p or even later for 7 days, but once I finish that week, I get "normal" 9ish to 5ish hours with the occasional overnight call during the rest of the month.
Many people automatically think that PICU attendings have a terrible lifestyle, but for me, I think it's a great spot. It's shift work in most spots, when I leave the hospital, I've handed my patients over and know they are cared for. There are risks that a sick or dying patient will extend my day, but if that happens, I feel an obligation to that patient and family to be there until things are at natural stopping point. And compared to my friends who are other sorts of specialists, they might generally work 9-5 and then have calls every 4th or 5th night from home, of which almost all will be at home. Some nights there are no calls, other nights, they may be up all night with calls from the ED, patients or other specialists (me!). That uncertainty to me is much more frustrating than having what many others would consider a "harder" schedule. Just something to think about.