Our taxable income this year was 34k. That's actually an excellent income for us, it tends to average 25-30k.
HOWEVER, about 10k of that was from investment income that is taxed because we pulled it out instead of reinvesting it. We have a lot of money in investments. What this means is that while we do live very VERY modestly and try (and succeed) to stay within our "income", we have little financial pressure. We can afford to buy our own health insurance. If we needed to, we could pull more stuff and pay off our mortgage within 48 hours (partially because we bought a modest home at a modest price and didn't get some crazy ass mortgage). If an emergency happened, we could deal with it. It would still be stressful, and we prefer not to touch the investments (they are, after all, going to put 3 kids through college/trade school/business startup and fund our old age), and there is a waiting period, but just knowing that that is THERE makes a HUGE difference. Even though we spend far less than most people we know with half or less of our net worth.
So I know that it is possible to do SAHMing (as long as you don't have a huge amount of consumer debt--we had none going into marriage) on a mid 20s to 30s income, I've been doing it for about 10 years now. However, don't underestimate stress, if it's really tight. In many respects I think knowing that I have $$ makes it easier for me to not spend money, and certainly makes me think about it less, though I think that is a bit counterintuitive.
You mention buying a "modest home at a modest price". The house next door to us is for sale. It's old, a little rundown (not a dump or anything), very modest sized, etc. The neighbourhood is nothing special (I like it, because we can get to a lot of things on foot within anywhere from 10 to 40 minutes...but the corner store and the local video place both closed in March). The yard is average sized, and average here isn't that big - when I saw the size of yards where my in-laws live, I about passed out.
The house is listing for $644,400. I checked. We'd love to buy it, and not have to move out of this neighbourhood. (FWIW, I grew up not too long from here, and it was considered a moderately crappy part of town - not really bad, or anything, but bordering on the "wrong side of the tracks". It has a slightly better rep these days, but it's really not a fabulous neighbourhood or anything.) There's just absolutely not way. We might have a prayer, on two professional incomes, but buying a "modest home" around here, on one income, is basically not possible.
Our rent, for a 1250 sq.ft. townhouse/rowhouse is over half of your taxable income. I really don't think we could even begin to manage on $34K. (Mind you, we do have more kids than average.)