No it doesn't. They're at sublight. I'm sure if they could have afforded zero-g they wouldn't have had artificial gravity either.The show fails to explain how they're getting to all the different planets.
Now how you have "dozens of planets with hundreds of moons" all habitable, terraformed or not? That's something else.
Again, Star Trek thinks that if you lose power you can fall into a sun in a matter of hours. Or fall past the moon and crash into San Fransisco in a matter of minutes. So cut Firefly some time/speed/and distance slack.