A recent experience brought this question to my mind: how problematic do you think it is for someone to have a "day job" and also serve as a priest?
As an ordained minister of the Church, you mean? That would depend on the day job and the responsibilities of ministry.
I don't see how it could work, in most cases. I respect those priests who do so out of necessity. But it seems like pastoral care, outreach, and daily ministry (such as Daily Office services or weekday Eucharist) would really suffer.
I've recently come across a number of priests and deacons who are also attorneys. I can see issues there.
Bi-vocational clergy are probably going to be more common as we go through tougher times for the Church and ordinary people's pocketbooks. Small congregations and many church plants simply won't have the resources to pay a full-time minister. There have been some Lawyer-Theologians in Church History. Depending on what type of law they practice, that could be a useful skill to have as a minister. However, I could imagine a corporate lawyer having a schedule way too busy to adequately pastor a flock (or properly do his job as a lawyer).
After Christianity became the religion of the Roman Empire, all Bishops worked in the civic-judicial process, pleading causes to higher-ups. hearing cases etc. You can find plenty of them complaining about having to deal with the paperwork, including Augustine. Didn't seem to hurt their spiritual life too much though. Maybe that's not a very helpful input. Isn't it still rather common in Eastern Europe for Orthdox clergy to combine that with some kind of other job?