The Verse In Numbers (aka TVIN) is an extensive fan work to make the Firefly universe internally consistent. It, and the Serenity RPG, are the excellent technical references to the Firefly universe. Are they canon? Here's what TVIN has to say about that in "Lies, Damn Lies & Canon".
...the next 100 pages (no, really, 100 pages) represent the most painstakingly-researched and thoughtful effort to make sense of a bunch of hooey that I've encountered...
So, what does that make The Map of the Verse? Canon? Extended Canon? Speculation?
Here's how I like to think about it - it's as accurate as it's possible to be right now. Meaning to say, it's 100 percent accurate, until Joss says it isn't. It's Joss' Verse, after all. We're just lucky that we were allowed to visit and bring home a few souvenirs.
The OP asks
So: how does she pull it off? TVIN describes a civilization with the ability to control gravity at a stellar scale. It mentions that quantum gravity has been solved in 2035 leading to extensive applications.
- The quantum nature of gravity is deduced, allowing for rapid and unprecedented advances in gravity manipulation technologies. Creation of artificial gravity and gravity screening soon follow.
TVIN goes a little bit into the history, but you're not going to get much more than something something quantum gravity.
...in 2006, a lab funded by the European Space Agency, created an artificial gravity pulse that measured 1/10,000th of a G. Unlike the previous claim, the experiments were successfully repeated, and gravity pulses were again created, but it wasn’t until the late 2020’s that artificial gravity fields were being produced of sufficient strength and duration to have commercial applications, and cheaply enough to be afforded by more than just governments.
By the time they arrive in The Verse they have such control over gravity they can compress brown dwarf stars to ignite them into small stars and use it to terraform worlds to have Earth-like gravity. This makes producing episodes cheaper.
Serenity manipulates gravity in two ways and TVIN has a whole section on the Gravity Drive. External gravity screening partially isolates the ship from the pull of gravity making her lighter. That's what the big rotor is on Serenity. Thus she needs less thrust to fly in atmosphere, reach escape velocity, and support her own weight.
In some ship designs, such as the Firefly design (all models) the primary gravity rotor assembly is visible as a rotating ring around the waist of the ship. The dampeners and rotor assembly work in concert to negate the force of gravity acting on the ship, and help dampen the effects of acceleration on the structure of the ship.
TVIN mentions an example where this is fundamental to the success of the Firefly class.
The hinge that allows the engines to fold down to parked and maintenance positions, is not strong enough to support the weight of the ship. Without gravity screening to reduce the weight of the ship, the engines would snap off of their mounts when lifting the ship.
Then there is the internal gravity field, that's what keeps the crew's feet on the floor. It also negates the effects of acceleration, though not perfectly. This allows the crew to be whipped around in extreme maneuvers for some dramatic tension.
Gravity field realignment easily compensates for typical maneuvering. However, sudden changes in direction and speed can temporarily exceed the rotors’ speed at realigning the internal gravity. In some cases, the onboard computer’s attempts at matching gravity alignment with existing natural gravity, during hard maneuvering, can actually magnify the adverse effect of sudden changes in acceleration3. This is why almost all lockers and cabinets on a ship have latches, and most shelves have some sort of guardrail or lip to keep items from sliding off.
Finally, the OP sensibly asks why gravity continues to work in "Out Of Gas" despite even life support being out. @Plutor speculates they would have extensive backups which is plausible if the ship will be crushed under its own weight or torn apart by maneuvers without it. Again, TVIN has something to say...
Without gravity screening, the engines would be directly connected to the axle, and it would be much thicker. With gravity screening, a firefly is quite nimble for a cargo transport. Without, it would be an ungainly hulk unable even able to lift itself while empty of any cargo. Additionally, the landing legs of a firefly cannot support the weight of the ship without gravity screening.
TVIN's explanation is that a gravity field only dissipates slowly after the generator is turned off...
However, an artificial gravity field is a series of pulses that occur very rapidly. The effect is continuous since the pulses are so close together that there is no discernable decrease between pulses. The longer the field is active, the longer the decay of a given pulse. After the field has been active for a few dozen hours, the decay between pulses virtually ceases to exist. Shutting down a gravity generator that has been in operation for months results in a decay of that last pulse that can last for weeks at near full strength.
Since according to TVIN "the only time the gravity screening is entirely powered down is while the ship is supported by a maintenance gantry", and we can be fairly sure Serenity hasn't seen one of those in a long time, we can assume Serenity in "Out Of Gas" has weeks of full gravity.