How long does grease take to go bad/rancid when sitting out?
Fat doesn't spoil like other foods. No bacteria can live in fat. Going rancid is a chemical reaction in which the fat molecules break down. So "leaving out" is not a problem.
The major factors in going rancid are light and air. The more light hits your fat, the sooner it goes rancid. Also, rancidity occurs when the fat is oxidized, meaning that if you prevent contact with air, your fat will last longer.
Both factors are minimized by transferring the fat to a tightly closed opaque container. With liquid oils, you use a dark colored bottle. As bottles are impractical for solid fat, just strain your grease into a jar, close it, and put it in a cupboard. The shelf life should be many months, probably more than a year.
Also, don't worry that you can get something nasty when the grease goes bad unnoticed. Rancid fat isn't a big safety risk, and ingesting it in small amounts is not problematic. If it stinks, throw it out. Before that, there is no problem eating it.
The above assumes pure, well-strained fat. Pieces of fried meat left over in the fat are a safety risk, even tiny ones.
Then what is so bad about rancid oils? If they can't infect you? Just it just taste unpleasant?
Actually, there is evidence of negative health effects in animal models, and there are also case reports of rancidity-caused illness in humans. See this article (citations included): http://nutritionnutsandbolts.com/2012/08/01/rancid-fat/