Know, oh prince, that between the years when the oceans drank Atlantis and the gleaming cities, and the years of the rise of the Sons of Aryas, there was an Age undreamed of, when shining kingdoms lay spread across the world like blue mantles beneath the stars - Nemedia, Ophir, Brythunia, Hyperborea, Zamora with its dark-haired women and towers of spider-haunted mystery, Zingara with its chivalry, Koth that borered on the pastoral lands of Shem, Stygia with its shadow-guarded tombs, Hyrkania whose riders wore steel and silk and gold. But the proudest kingdom of the world was Aquilonia, reigning supreme in the dreaming west. Hither came Conan, the Cimmerian, black haired, sullen-eyed, sword in hand, a thief, a reaver, a slayer, with gigantic melancholies and gigantic mirth, to tread the jeweled thrones of the Earth under his sandaled feet.
The very short answer is, there is no such period. The Phoenix on the Sword, the first adventure of Conan the Cimmerian, began life as a revision of a story of one of Robert E. Howard's previous series characters, Kull of Atlantis, which were much more explicitly set in a fantasy, sword & sorcery milieu. As Patrice Louinet traces the origin and development of the Conan stories in his essay "Hyborian Genesis" in The Coming of Conan, Howard spent a considerable amount of time developing a consistent background for the Conan tales - including drawing a full map and going through four drafts of a lengthy pseudo-historical essay entitled The Hyborian Age.
According to Howard, the adventures of Conan are set in a period before contemporary history, but after the Younger Dryas. As Howard put it in his essay:
When I began writing the Conan stories a few years ago, I prepared this 'history' of his age and the peoples of that age, in order to lend him and his sagas a greater aspect of realness. And I found that by adhering to the 'facts' and spirit of that history, in writing the stories, it was easier to visualize (and therefore to present) him as a real flesh-and-blood character rather than a ready-made product. In writing about him and his adventures in the various kingdoms of his Age, I have never violated the 'facts' or spirit of the 'history' here set down, but have followed the lines of that history as closely as the writer of actual historical-fiction follows the lines of actual history. I have used this 'history' as a guide in all the stories in this series that I have written.
Many of the names of nations, cities, peoples, and individuals were borrowed from Howard's reading in Bullfinch's Mythology and other sources; the end result was a rather eclectic mix that belonged to no one period, but referenced many different periods. As Lovecraft put it:
[...] Howard has the most magnificent sense of the drama of “history” of anyone I know. He possess a panoramic vision which takes in the evolution & interaction of races & nations over vast periods of time, & gives one the same large-scale excitement which (with even vaster scope) is furnished by things like Stapledon’s “Last & First Men”. The only flaw in this stuff is REH’s incurable tendency to devise names too closely resembling actual names of ancient history—names which, for us, have a very different set of associations. In many cases he does this designedly—on the theory that the familiar names descend from the fabulous realms he describes—but such a design is invalidated by the fact that we clearly know the etymology of many of the historic terms, hence cannot accept the pedigree he suggests. Price & I have both argued with Two-Gun on this point, but we make no headway whatsoever. The only thing to do is to accept the nomenclature as he gives it, wink at the weak spots, & be damned thankful that we can get such vivid artificial legendry.
The result is thus very ahistorical and anachronistic: steel weapons and writing before such things were invented, 17th-century buccaneers alongside a quasi-Roman Empire alongside a strip of frontier that could have come from Texas in the mid-1800s, Egypt under the Pharaohs, etc... However, this gave Howard great leeway to basically tell whatever kind of story he wanted - and did. The Conan tales are extremely diverse in terms of the genre and style, and the settings of those stories in part reflect the different historical periods he had interest in. So for example, in "The Phoenix on the Sword" Conan's position as a "barbarian" king taking charge of a "civilized" Aquilonia recalls Odoacer declaring himself King of Italy in 476; "Beyond the Black River" is essentially a frontier-tale, with the Picts taking the place of Native Americans; "The People of the Black Circle" echoes border-conflicts between the Sasanian Empire and various Central Asian peoples, etc.