I've always this was a no-no, but can't I leave some stock simmering on the lowest of low heats and do some errands without risking burning

Leave it in a crockpot!

In the professional kitchens I've worked in, we have always started the stock to simmer just before leaving for the night. My biggest concern with it for around the house would be the subsequent gas or electric bill (unless I had animals or children that is). At home, I make my stock in a pressure cooker. It comes out good and takes a fraction of the time and effort. Add all ingredients into the stock pot. Bring it to pressure. Cook at pressure for 30 min. and let it cool on it's own accord.

My disaster: getting ready to leave for work, I put dry beans and water in a pot, put it on high heat and left the house. I was supposed to bring it to a boil and turn it off (nothing cooks when I'm not home!). It was almost lunch time when I remembered. I rushed home to find a house full of smoke and my pot melting and fusing with the electric burner. The beans were carbon. Fortunately, no fire, no harm other than the pot -- total loss; smoke everywhere; dinner would not be chili. The only way to try this safely is with a crock pot.

Could you warm up the oven, shut it off, and leave the stock in there?

I did it and burned my kitchen came home to firetrucks at my house, I had errands to run, pick up the daughter from soccer practice etc... its risky.

I'm on the fence, but since I have a 10 year old too I think I'll err on the safe side . Granted my 10 year old is a yellow lab, but who knows what kind of crazy she'll get up to with the smells of chicken stock wafting through the house.

I just did it yesterday. I made sure that it was up to temperature and on the lowest of simmers, added a little extra liquid and did not leave a 10-year-old in the house. Worked fine for me. On the other hand, I know I've heard that the best way to burn down your house is to put a pot of oil on the stove for French fries and then go out to buy your potatoes.

I agree. My mom once left a pot of soup on very low heat while she and I went off to run a quick errand. My brother was there to "keep an eye on it," but he was 10 and playing a video game. The house was filled with smoke when we got back. "Low heat" turned out to be "high heat" by accident. I believe my mom threw the pot out with dinner and started over.

Would not advise it, Murphy's law invariably kicks in, & an unwatched pot tends to boil sooner than usual!