Depends what, for these purposes, is a "planet".
In Firefly, they don't really give details. The best description I've seen has it as a multiple-star system, in which each star has planets, and many of the planets have multiple moons - many of the moons and planets were of a size for terraforming. That way, getting into the tens or dozens of worlds isn't too implausible.
Without hunting down facts, this sounds about right with the representation of the setting, having just finished re-re-watching the series last night.
In at least on episode, they referenced that they were on a moon. So likely some of the gas giants had viable candidates.
Near as I can tell, The Firefly premise is:
earth got over-populated/resources got scarce
a distant star system was identified and terraformers were sent and then a mass exodus occurred from Earth.
Earth blows up (referred to as the "Earth that was" in the intro to Serenity if I recall). Quite possibly what they really mean by "used up"
space travel appears to be sub-light, but faster than NASA current tech, and other than being in a multi-star system, is limited to that solar system
For Firefly:
given the assumed lack of FTL, my guess is they identified a star system with lots of planets
They send robot terraformers, at new tech sub-light speeds, that took a few hundred years
they sent sleep/generational ships behind the terraformers, schedule to arrive shortly after terraforming completed.
Not all terraforming is equal, so some planets suck. if your colony ship drew Planet#12, you were stuck with a frozen iceball until interplanetary travel picked up
New inter-planetary ships get built (military, freighters, passenger, etc) to bring people/stuff from the inner/outer planets)
Ships like the firefly class get built, used, retired
Earth finally dies.
Barring specific years defined in the show, it's feasible that the setting for Firefly is about the new society that forms when man gets out of the Sol System because Earth or the Sol itself are due to expire finally. So be it 2500AD or 3 million years in the future, the effect is the same. You can't go to Earth because it's gone.
Now a sly trick is like what happened in Asimov'sFoundation series, Earth is there, it's just been forgotten about and lost.
Now for the OP's setting, if you're going to say there was a big multi-star network for space travel and then say it's wrecked, then the players are pretty much stuck to playing in a single solar system anyway.
Something to realize, as it's a pet peeve of mine when the GM sets something out and then negates it, then why bother anyway...