In mainstream physics, which comes to us via Einstein, it is believed that nothing in the universe can travel faster than the speed of light.
If this is true, the distances between bodies in space are such that travel between and among stars will always exceed any human's lifetime, usually by many orders of magnitude (leaving aside time-dilation for the time being.) For this reason, many science fiction stories (including all of those in Star Trek) have used faster-than-light travel as a plot device. But no one disagrees that this plot device, at this time, appears contrary to reality, and one of the things that sets science fiction apart from other genres is that it maintains some level of concurrence with reality as we understand it.
For these reasons, it would be a bold move dramatically to posit a universe in which the Einsteinian view prevails, since this limits the dramatic options significantly. The world of Firefly does appear to be consistent with the Einstein view. The issue has been (and in muted fashion does continue to be) a matter of hot debate, but statements from the creator Joss Whedon, from the Serenity Movie, and the general look of the TV episodes all combine to suggest that the light-speed limit is honored in the world of Firefly.
A number of things flow from assuming no faster-than-light travel. Most importantly, there is no transit among stars, only among planets and in a single solar system. This precise arrangement was laid out specifically in the opening moments of the movie Serenity.
The two big arguments suggesting this is false are: (1) Even getting to the nearest star is nearly impossible without light speed, and Firefly requires that at least some group got from here (Earth-that-was) to there (Firefly Universe), which is not necessarily even a close star, and (2) Communications take place instantly in Firefly. Even between planets (for example Earth to Mars,) there would be a significant delay between each question and answer (about 20 minutes if Earth to Mars.)
The answers to these questions are simply (1) "nearly impossible," of course, means "possible", plus the storyline gives us 500 years to play with, and (2) the dramatic flow of the story can tolerate being confined to a single solar system, but having no real-time communications is simply too great a dramatic limitation, so it is swept aside.
That last is bad, definitely, but the fact remains that in this sort of science fiction, less hand-waving is still better than more hand-waving. Firefly makes no pretense to be consistent with all actual scientific theory ("artificial gravity" is a particular sore-spot for purists) and the creator Joss Whedon has disclaimed any responsibility for scientific fidelity. Nevertheless, Firefly is truer overall to scientific principles than nearly any other comparable production.
The prospect of any particle of any mass reaching light speed is precluded by modern physics, but only a bold author would have the courage and skill to make a good story and honor this (so-far) good science at the same time. From all appearances, Joss Whedon has done this, to excellent effect, in Firefly.