"Two-handed Engine", a novelette by Henry Kuttner and C. L. Moore; first published in The Magazine of Fantasy and Science Fiction, August 1955, available at the Internet Archive.
Expository paragraphs from the beginning of the story:
Ever since the days of Orestes there have been men with Furies following them. It wasn't until the Twenty-Second Century that mankind made itself a set of real
Furies, out of steel. Mankind had been through a lot by then. They had a good reason for building man-shaped Furies that would dog the footsteps of all men who kill men. Nobody else. There was by then no other crime of any importance.
It worked very simply. Without warning, a man who thought himself safe would suddenly hear the steady footfalls behind him. He would turn and see the two-handed engine walking toward him, shaped like a man of steel, and more incorruptible than any man not made of steel could be. Only then would the murderer know he had been tried and condemned by the omniscient electronic minds that knew society as no human mind could ever know it.
For the rest of his days, the man would hear those footsteps behind him. A moving jail with invisible bars that shut him off from the world. Never in life would he be alone again. And one day—he never knew when—the jailer would turn executioner.
A description of the Fury:
The Fury was taller than a man. It stood there for a moment, the afternoon sun striking a blinding spot of brightness from its shoulder. It had no face, but it seemed to scan the restaurant leisurely, table by table. Then it stepped in under the doorframe and the sun-spot slid away and it was like a tall man encased in steel, walking slowly between the tables.