Sleep on a clothesline
Sleep very soundly.
To say that one feels tired enough to sleep on a clothesline seems like a mere fanciful metaphorical way of saying one is extremely sleepy. No one would expect you to actually balance on a line and the idea that the phrase originated from such a literal event sets the folk-etymology alarm ringing full pelt. Nevertheless, there are well attested stories that, during the depression of the 1930s, destitute men did spent their nights hanging over ropes which were strung across rooms for them to sleep on.
In Down and Out in Paris and London, 1933, George Orwell recorded a London sleeping establishment, known as The Twopenny Hangover, with just such an arrangement:
At the Twopenny Hangover, the lodgers sit in a row on a bench; there is a rope in front of them, and they lean on this as though leaning over a fence. A man, humorously called the valet, cuts the rope at five in the morning. I have never been there myself, but Bozo had been there often. I asked him whether anyone could possibly sleep in such an attitude, and he said that it was more comfortable than it sounded - at any rate, better than bare floor. There are similar shelters in Paris, but the charge there is only twenty-five centimes (a halfpenny) instead of twopence.
(The attached photograph is claimed to be of an American institution from the same period. I have no means of verifying its authenticity.)
The expression 'sleep on a clothesline' is itself quite esoteric and probably the best reason for including it here is to refute the commonly expressed suggestions that such sleeping habits form the derivation for the word 'hangover' and the expression 'sleep tight'. Neither of these is supported by any evidence.