I am worried about the safety of the broth
Politically correct answer is no. Reality? We do it all the time at our house, and have never had any issues...
If it was warm when you went to bed and had a lid on it, go for it. I am with Jane - just not that paranoid. Two days? No. Just get it in the fridge/freezer this morning and you are good to go.
My motto is "If in doubt, throw it out" having had a bout of food poisoning a few years ago (not from anything I cooked) I err on the side of caution always. It's always possible as Jane and Susan said that it would be fine but me personally I never take chances.
I'm not a food scientist, but I have had food poisoning. (self inflicted. let the beef stew cool too long before refrigerating. really awful. seriously dreadfully awful.) Here's what the USDA says, "Bacteria grow most rapidly in the range of temperatures between 40 and 140 °F, some doubling in number in as little as 20 minutes. Some types will produce toxins that are not destroyed by cooking." Soyou can kill the bacteria. But you can't remove the toxins, and it's the toxins that can make you wish you were dead. There's no way to know if there are toxins present. I've done the same thing and had to throw out what seemed like perfectly good chicken stock. But I never (ever) want to get food poisoning again.
If cooking for myself, I'd probably use it - for others, no. But I have a pretty strong stomach.
Having said that: Michael Ruhlman's gone on record about leaving stock out on the stove top overnight to cool (in fact, up to 3 days)
And then subsequently further clarified with this post after discussion with McGee that it's not bacteria but spores that are the issue:
If you'd ever had food poisoning, you wouldn't be asking this question. And actually, my rule of thumb is "if you have to ask, the answer is no. I vote for dumping it.
I have found that chicken stock goes bad pretty quickly. Having had food poisoning, I say pitch it.
I do it all the time and have never had a problem - note that I am not a long sleeper - turning off the stove around midnight and waking at 7 - so it's really not out for that long, I guess.
I am a physician. Drbabs and Sarah Reinertsen both make important points - reboiling the stock will not eliminate any bacterial toxins which could make you sick, and spores are also a big problem (and may not be eliminated by re-boiling either).
I'm with all the other people who have had for poisoning. It's bad enough to really make you think twice about food that has been sitting out. I'm not one to be timid about food. I eat stinky cheeses, drink raw milk (in case you were wondering, it's never made me sick), love sushi, and am not squeamish. But when it comes to common sense food safety, I vote for better safe than sorry.
The above example of Michael Ruhlman keeping stock on the back of the stove for a week is horrifying. Stock is the perfect petri dish for bacteria, and boiling it may make it "safe" but will it really taste good after a week of doing that over and over? I think Harold McGee also said in the above article that when you reboil stock over and over, you're just killing millions (perhaps billions) of bacteria, which will then flavor the stock with their dead bodies. Yum.
Here's the method I use for cooling just made stock. By doing this I avoid the temptation to leave the stock on the stove overnight. Carefully pour the hot stock into a large bowl. Fill a plastic storage container that is small enough to fit in the bowl easily, with ice cubes. Put the ice filled container on top of the hot stock, being careful not to let stock get into the container. Let it float there til the ice melts - it usually take only a few minutes. You may need to repeat the process a time or two, until the stock is room temp, then store as usual - fridge or freezer. The whole process takes around 15 minutes and works like a charm.
I have had food poisoning several times (travel related, never at home) and I would still just boil it again (so long as it was just stock). Maybe I am just reckless, but I have done this at home several times to no ill effect.
I left a store bought carton of partially consumed beef bone broth on the counter overnight and have the same conundrum. Also seeking an answer, I have a couple of questions:
What's the difference from between an unrefrigerated carton of broth that had been sitting on a grocery shelf unopened for months and that same item partially used but uncapped for under a minute then left out unrefrigerated for 8 hours? Has the load of "toxins" or spores increased that much that bringing it to a boil to kill harmful bacteria would cause it to be unquestionably deleterious to one's health if consumed?
I'm keeping the remaining 16 oz in the refrigerator awaiting your response.
If this happens to you again, I'll tell you our trick that we use in the winter. We just did this yesterday. Cooked a lovely stock and we were too tired to wait for it to cool and the fridge had no room for a large stock pot. We have an enclosed front porch so we strained it and put it out there to cool. Unfortunately, it's fridge temp here now. Tonight we slipped the fat off the top and froze it. If you have a cold room, it might be a good fix.
Brief (~5-10 min) boiling in closed container kills bugs and spores and inactivates toxins, eg botulinum and shigella toxins. If container is closed, where do new bugs come from? Keep it covered, and reboil it in the morning before cooling and freezing.