I have to say, this song certainly seems to be trying to tell a story. But it's mostly told in the spaces between the words.
What sort of establishment is the House of the Rising Sun? The conventional wisdom is that it was a bordello. There was indeed a noted brothel in New Orleans by the name Soleil Levant (French for "Rising Sun"), for a dozen years from the time of the Union occupation during the Civil War. After it was closed, it lived on in a folk song, which was rocked-out by the Animals a century later.
Or maybe it was a women's prison? There was one not far from New Orleans with a painting of a sunrise on its gates.