Depending on what type of cases you do, your patients will generally be sick, in the ICU (at least for a few days), with all the potential for trouble there. However, they aren't "liver transplant" sick.
Thus, you can expect more middle of the night phone calls from your residents (Surg Onc is pretty academic), rounds on weekends, etc. But there are *few* emergencies - ie, you won't be in the ED seeing a pancreatic cancer pt needing a emergent operation (unless its one of your patients with a complication).
It totally depends on what you want. You focus on breast/melanoma/etc., you can have a nearly entirely outpatient practice. You do only esophagus, hepatobiliary and pancreas you end up with a big list of people that can be quite ill.
Likewise, it depends on whether you have residents (I do, and my pager goes off literally maybe 2-3 times a day, and maybe 1-2 times/week after 5PM on weeks that I'm not on call for the ED), or share call in a complete way with your partners, but assuming that you want to answer ALL you own calls by being in a solo private practice, the number of calls will be determined by how well you tuck your patients. During my fellowship when at the private hospitals, I often got <6 pages/day even when doing the HPB service because I'm good at rounding and tucking people in and our nurses were good about not bugging you unless they really needed you.
For my non-ED call, it is very unusual to need to come in to evaluate a patient in the middle of the night. I do get transfers from other hospitals, and you get the occasional call to the OR to help out with a big case that someone else has gotten themselves into trouble with, but overall not onerous.
In short, I would not consider lifestyle to be an obstacle in your average surg oncologist's decision making to entering the specialty