I looked over the Science Fiction Encyclopedia page about Space Opera and found a candidate which beats Asimov's Foundation series in publication date by just two months: the story "Recruiting Station" by A. E. van Vogt, published in the March 1942 issue of Astounding (Asimov's first Foundation story was published in the May 1942 issue of Astounding). I have a collection of Astounding issues on CD-ROM (they're available from this ebay seller if anyone's interested) so I was able to look it over. The story involves two characters from the twentieth century (the story starts in 1941), "Norma" and "Garson", who are forced to work for a time-traveling empire from the distant future called the "Glorious", who are a sort of totalitarian state of enslavers and brain-washers, and who are at war with the more humanistic "Planetarians". The two civilizations are based on different worlds, though still all confined to our Solar System (but you did mention Firefly which was also confined to a single system). On page 19 an agent of the Glorious explains some of the history:
he said quietly: "Several hundred years ago, a mixed commission of Glorious and Planetarians surveyed the entire physical resources of the Solar System. Men had made themselves practically immortal; theoretically, this body of mine will last a million years, barring major accidents. It was decided available resources would maintain ten million men on Earth, ten million on Venus, five million on Mars and ten million altogether on the moons of Jupiter for one million years at the then existing high standard of consumption, roughly amounting to about four million dollars a year per person at 1941 values.
"If in the meantime Man conquered the stars, all these figures were subject to revision, though then, as now, the latter possibility was considered as remote as the stars themselves. Under examination, the problem, so apparently simple, has shown itself intricate beyond the scope of our mathematics.
Later on p. 22, this character reveals the war is interplanetary:
We must win; our cause is overwhelmingly just; we are Earth against all the planets; Earth protecting herself against the aggression of a combination of enemies armed as no powers in all time have ever been armed. We have the highest moral right to draw on the men of Earth of every century to defend their planet.
Most of the action of the story takes place on Earth (present and future), so I'm not sure if this story really meets the condition of being the "first story set in space" with multiple civilizations and no aliens. But it might be seen as meeting that condition, since there is one part of the story where Garson manages to escape the Glorious and is taken on a Planetarian spaceship heading to Venus, where he meets characters from various other times (including other civilizations from times earlier than the war but still in our distant future) who have been recruited to fight in the war by the Planetarians. He gets close enough to see Venus looming large in the window, but is then yanked through space and time by Norma, who has been granted psychic powers by descendants of humanity from even further in the future, the "four hundred and ninetieth century A. D." who are "human in name only". Still, I can verify that there are no actual aliens in the story, everyone is some sort of descendant of modern humans (along with some intelligent computers).
Incidentally, there are also some earlier stories that are set centuries in the future, and which feature civilizations on other worlds of our solar system, where the inhabitants of these other worlds seem to be entirely human-looking. I would say these otherworldly denizens are probably intended to be aliens who just look like humans as in the old Buck Rogers and Flash Gordon comics (or later works like Star Trek), but perhaps there is a little ambiguity as to whether they could be descendants of human colonists. One example I found that seem to fall into this category--though I only read plot summaries--is Hugo Gernsback's Ralph 124C 41+ which originally appeared as a magazine serial beginning in 1911, which is set in the year 2260 and features a romantic rivalry between an Earthling and a Martian named Llysanorh' CK 1618 (the summary here notes that 'the planets are populated by more or less human populations who can interbreed, Martians being tall and large-chested'). Another is Ray Cummings' Tarrano the Conqueror from 1930 (available online in its entirety here, though I only skimmed it and looked at the illustrations) which is set in 2430 and features a Napoleon-like Venusian villain who starts a war between planets. Cummings also wrote another story of warring planets that was published in 1930, Brigands of the Moon, which was set in the year 2079 and which also featured Martians and Venusians, and according to the summary here, "Martians are humanoid, but gigantic; Venusians are petite and sexy; both apparently can interbreed with Earthmen." So both the relatively near-future date and the physical differences suggest they were probably intended to be very human-like aliens, which makes it more likely this was the intent in Tarrano the Conqueror as well, though the summary of Tarrano here, by the same author of the other two summaries I quoted, refers more ambiguously to "Venus and Mars, the natives of which are human or humanoid".