I'm a vet tech and have to deal with that kind of stuff all the time. Sometimes we can go for weeks without a euthanasia, and sometimes we'll get 3 or 4 in a week. It's really hard at first, but you have to learn to separate yourself somewhat from it. We definately feel for the clients and are sympathetic and give hugs and stuff, but if we were to break down and cry, it would just upset the owner even more and make them feel like a bad person for doing it.
The hardest ones to deal with are the clients that you've really grown to know. Last week one of our favorite clients had to put his dog to sleep. It was really hard. We all feel so bad for him and we are still thinking about how sad it is.
But you cannot dwell on this side of the job because there is so much more to it, so many happy things about the job!