There are many Catholics today who claim that no priest can take care of a family and a parish at the same time; they believe it is just too much for one person to balance. Not trying to draw attention to myself, but I think I can contribute at least a little bit to the discussion. For those of you who are unfamiliar with my situation I will give a brief summary. My wife and I have been married almost 30 years now, and we have five children (oldest is 23, youngest is 9). I was given special permission to be ordained as a priest by Pope Benedict XVI seven years ago. Prior to my reception in the Church, I served for sixteen years as a protestant minister, so my wife and children are all familiar with what it means for me to work in a ministerial setting.
Now, I must admit, right at the start that I am busier now than I ever was as a protestant pastor, but that only goes to support the point I am making. Therefore, I will say that it is possible for a married Catholic priest, not just to get by, but to maintain a good marriage, have all five of his children stay in the faith, and maintain a pretty good relationship with the members of my parishes (I serve three right now). This does not mean that we have never experienced any problems; both my family and my parishes have experienced their own share of challenges. It does mean, however, that it is not impossible for a married Catholic priest to attend to his duties with both his own family and his parishes.
Having said that I must say, it is very, very hard to do it right. In fact, if the option was open, and a young man came to me trying to discern whether to get married before pursuing the priesthood, I would tell him, "No, choose the celibate life instead." Yes, that might shock some people, but it is true. The current custom of priestly celibacy is definitely best, and it does enable the man to attend to his priestly duties in a clearer fashion (as St. Paul says, cf. 1 Corinthians 7:32-34). If God genuinely calls a man to do both vocations of husband and priest (and it is my opinion that this happens less often than many people think), then He will grant him the gifts to do both. Not every man is called to celibacy, nor is every man called to marriage (cf. the important comment of St. Paul in 1 Corinthians 7:7!).
It is difficult to be a good Catholic father to any household; if you add in a ministerial responsibility that can call you away at a moment's notice (24 hours a day), then it makes it even more difficult. As I said, "difficult", but not impossible. This is why I will say (and have said many times), I am not an advocate of eliminating the custom of priestly celibacy. Some men (and I really mean only "some") can maintain a good marriage and keep their children in the faith. Few priests (and I really mean only "few") can maintain a good marriage and keep their children in the faith. Open the doors up to large numbers of married priests in the Catholic Church (especially here in America), and you will be opening up the doors to a whole new set of problems that were never considered.
I will even mention (in just general terms) that my family has experienced a few "trials and tribulations" but we have come out faithful and strong in spite of them--due largely to the fact that we raised our children to work through their problems with the help of the rest of the family. In addition, my parishes have had some experiences that were difficult (which one has not?). Yet, none of these challenges had anything to do with the fact that I am married with children still in the home. In some ways my experience of working directly with my own family helped me to know better how to lead the parish through its challenges; but that is not necessary for every priest.
One thing that people do not think about is that in some ways the priest frequently treats his own family as he does other parish families --- he needs to minister to them according to what they need. Someone once asked the direct question, "how do you choose between family and parish when the two conflict?" Even though it may seem like that is a genuine conflict that celibate priests never experience, it is not true. There are times when I have had to leave a meeting with a parish family because I got called to something more serious (like giving last rites); they always understood. There are times when I have to leave when I am with my own family (like the time I had to leave my youngest boy's birthday party because of a ministerial responsibility--and he completely understood, and held no grudge over it).
If a married priest is really doing the (additional) hard work to maintain a good marriage and if his wife is supportive of him in the raising of their children, then their family bond will be such that the wife and children understand what it means to "share" him with an entire parish, and he will never have to neglect either. As I said above, however, it is not easy to accomplish this. If the desire is there to allow more married men to become priests, then the vetting process will need to become twice as stringent as it is now. This is precisely because the responsibility of a married priest is more challenging (but, once again, not impossible).
Catholics really need to stop using the "it's impossible" argument against extending the option of married priests. It is not accurate, and it is far too pragmatic (and pragmatism is rarely a good rule of thumb for determining what is best in the Church). Just because something is difficult does not make it wrong. It is difficult for me to celebrate 8 Masses in 72 hours, but I do it every Easter season -- because it is good and right. It is difficult for parents to raise their children properly in our modern pagan society; that does not make it wrong. There are plenty of reasons not to allow for more married Catholic priests, but "its impossible" is not one of them.