All those people we see in the background, not wearing uniforms are likely citizens...
Weird question, I think. Trillions across the Federation sitting around at home while the few serve in Starfleet....?? no
Probably more likely in the 100 billion range, tops.
I quote from DS9 "Statistical Probabilities":
BASHIR: It's inescapable. There's no way the Federation is going to be able beat the Dominion. We have no choice. We're going to have to surrender.
[Captain's office]
SISKO: Surrender to the Dominion. Not on my watch.
BASHIR: Sir, I understand how you feel. I don't like it any more than you do, but it's the best option. We've run dozens of different scenarios. Even if something unlikely were to happen tilting the scales in our favour, such as an anti-Dominion coup on Cardassia, we'll still lose this war.
SISKO: But that doesn't mean we should just give up and roll over.
BASHIR: If we fight, there will be over nine hundred billion casualties. If we surrender, no one dies. Either way we're in for five generations of Dominion rule. Eventually a rebellion will form, centring on Earth. It'll spread, and within another generation, they'll succeed in conquering the Dominion. The Alpha Quadrant will unite and a new, stronger Federation will rule for thousands of years. Since we can't win this war, why don't we save as many lives as we can? I know it's difficult to accept.
So if the Federation is projected to have over nine hundred billion casualties during the Dominion War, it should have a population considerably more then one hundred billion.