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Being in a desperate or bleak situation.
It's said the origins of this phrase derive from boxing, where being on the ropes is considered a bad place to be. Why? When a boxer is backed into a corner or is up against the ropes, their movement is severely restricted, which makes dodging incoming punches much more of a challenge. Thus, a boxer who is up against the ropes is in a dangerous or desperate situation.
Additionally, a boxer who receives a series of heavy punches may attempt to grab onto the ropes to prevent himself from falling over, hence, he would literally be on the ropes and in a tough spot. A boxer never wants to be on or even near the ropes, because it usually means he's in a lot of trouble. It's easy to see, then, how this boxing term could be turned into a phrase for anyone that finds themselves in a difficult situation.
The saying goes back to at least the year 1820, where it's recorded in Blackwood's Magazine. The phrase is used in an article about boxing, titled Boxiana, where it reads:
"At the close of one round, when Bill had got his adversary on the ropes, he went over him in a
summerset, in a way that we do not not remember to have seen practised either before or since."
* Tom was sick and with no food in the house to nourish himself, he was really on the ropes, that is, until he decided to drop by the grocery store for some canned soup.
Note: The origins for most idioms are unclear. Often times, the origins you see listed are plausible theories to how an idiom came to be, but not necessarily so. The quotes you see that contain the phrase are the oldest that I could find, but it's very possible there are older recordings somewhere, so if you know of any, let me know!
Keep in mind, just because you see a saying in a newspaper from 1850 does not mean it originated in that year, or from that newspaper. In all likelihood, if a saying is already being used in a form of media like that, it's probably from an earlier time. The purpose of these old quotes is to show, with proof, how old some phrases go back in history.
Know Your Phrase - Common Idioms and Phrases, Their Meanings and Origins > O-Letter Sayings