On this page of the Disco Elysium guide, you will find an accurate description of completing the Getting the Hanged Man's boots side task.
In the Getting the Hanged Man's boots task in Disco Elysium, you must get the ceramic doghouse that you found on the feet of a hanged man.
You will receive the task from Call Me Manana, but only after talking to Cuno about the remaining parts of ceramic armour (the Where is the Rest of the Armour task? task).
Once you're left alone, head to the place where you left the corpse - the yard behind the Whirling-In-Rags building or the Doomed Commercial Area if you put it in the fridge (the Fridge victim's body task). When you try to get the boots, they will come down with pieces of flesh, so you will need to clean them.
Go to the kitchen at the Whirling-In-Rags Hotel and interact with a large pot on the gas stove. Add soap and white vinegar inside, then boil your shoes to clean them and get rid of the nasty smell. When the cooking process is over, Fairweather T-500 Greaves ceramic shoes will arrive in your inventory.