I have been Orthodox for 12 years. I have never seen or heard of any *really* large families in all that time. I have seen a few with six kids, which to me is still quite manageable.
Just sharing my personal experience:
We have four children. During the time they were coming along, we used no contraception. They all arrived roughly 2 years apart.
During my last pregnancy, about halfway through, I began having extremely strong thoughts and feelings of "this is the last time I can do this." Too strong to ignore. After she was born, and after much prayer and counselling with our priest, we made the decision for my husband to have a vasectomy (I would have had a tubal after the c-section, but it was a Catholic hospital and they don't perform them).
A few years after that, I developed problems that necessitated having the Novasure procedure done. Pregnancy after this procedure is extremely dangerous and is to be avoided. So, he would have ended up having the vasectomy anyway, eventually.
As far as affording kids, my first thought is that it is not NEARLY as expensive to raise a child in this country as the media would have us believe, particularly if one parent can stay at home and avoid childcare costs. We lived solely on one income from 1995-2009 and we've been fine.