I read this story when I was a kid in the early 1960s. I don't remember the plot, but I was struck by the idea of this market where vendors wanted to buy (and sell?) things (namely pieces of human bodies, it seems, like single strands of hair, fingernail clippings, etc). My recollection is that the story had a whimsical feel to it, but I may be conflating that with some other story.
The date doesn't match your recollection, but otherwise your description sounds like "The Skinny People of Leptophlebo Street", a 1975 short story by R. A. Lafferty, which was also the (unaccepted) answer to the old question Story where man goes to borrow money from odd community of hoarders. It was published in the original-stories anthology Orbit 16 (Damon Knight, ed.), which can be borrowed (free but registration required) from the Internet Archive.
The story seems to be set on Earth; at any rate, there is no mention of planets or space travel. Also, there is no explicit mention of fingernail clippings.
"Lose weight free in seven-minute surgery, sir," a small boy chanted. "My father does good free work. He is one good loser."
"No, not right now, boy," Canute said.
"Have your appendix out, sir? Have your appendix out?" another small boy was putting the shill on. "My father performs faithful appendectomies free."
"No, not right now," Canute said.
"This boy is Pat Thingruel, the brother of Piet and the son of Jan Thingruel," Effie told Canute. "His father is as stylish a free appendectomist as you will find anywhere."
"I do not understand how all the people of Leptophlebo Street can work for free," Canute said. "How do they profit by it?"
A second curious grackle-bird came down and got itself squawkishly stuck in the bird-lime of the edging of Effie's garden-hat.
"Oh, there's a lot of profit!" Effie exclaimed. "A vermiform appendix, especially when inflamed, is a veritable storehouse of richness. Master microchemists like ourselves can manufacture all sorts of useful things from such rich material. And the teeth that Royal Shortribs cleans, do you realize how super-organic are the deposits taken from teeth? Do you know how many things can be woven and fabricated from the hair that Claude Halfgram cuts? Garments, rugs, tents, seines, modish gowns for the modish ladies in the town. Almost solid profit. And the head-grooming that he does, do you know that there are some very lively products to be had from that? Our greatest industry, though, is the night soil that we gather from the cooperative people of the town. And I will tell you something else if you will promise not to tell the monkeys."
"No, I won't breathe a word of your secret to the monkeys," Canute promised.
"We pay the monkeys only half as much per equal weight for their night soil as we pay the people in the town. And the monkeys bring theirs to us; we don't have to go and get it. Ah, there is profit everywhere you look, in the stones, in the air, in the very rain. What a money-harvest we do have! Mills and mills and cents and cents, and at the end of a week we may even have another nickel for our hard work."
"It's a wonder you don't gather belly button fuzz and process it for profit," Canute laughed.
"Of course we do," Effie cried. "We gather more than a pound a year of it from the people of the town, and this in spite of the fact that many of the burghers will not cooperate with us and say that the whole thing is silly. But there are a few friendly people in town who wear wool. The woolies are the best for the fuzz. And it can be made into the softest of all sheens. Oh, do stay over and have a look at our night-life tonight, Mr. Freeboard! Really, it's wonderful the times that we do have."