First of all, pardon the typo as I meant to type: "compulsory service" but you can't edit rushed titles :P.
I know the answer is "no." But... with "death lists" longer than an entire space station of people can count... how in the hell does Starfleet maintain recruitment during this time? Were there defectors? I know they mention "personnel shortages" but you have to wonder... why didn't Jake enlist during the war? I have logistical questions about how they maintained the human resources to keep this war going in, what is actually, a freerider society where you can employ yourself as a "bad artist" and have every individual need met to your own utopian paradise.
Why would they need conscription? I imagine there were plenty of recruits. After all, the Dominion wished to conquer them. It wasn't a border dispute or anything petty like that; they were being targeted for full-on invasion and conquest. You'll never run out of people willing to defend their homes by fighting in these circumstances. I imagine the harder the war got, the more people you had lined up to enlist. When Betazed fell, for example, I imagine floods of citizens on neighbouring Federation worlds went out to join the war effort.
Another important point is that the Federation is a free society. It is
worth defending, worth risking one's life for. Forced conscription would undermine that. I for one would never fight for any nation that employed conscription, for a nation that refuses to accept my freedom and views me as, essentially, a slave. Where's the point in fighting to protect your children if the government views your children as simply further potential soldiers, o be placed in harms way themselves at the first opportunity? I imagine those fighting for the Federation fought for a love of freedom, not out of any form of primitive nationalism. Forced service is the Dominion and Cardassian way, not the way of the Federation.
You say, why didn't Jake enlist? Your reasoning appears to be, "well, he's young and healthy, so he should fight". That is not a very sensible view. Such thinking is what mires a people in conflict. When you have the expectation that your young people should be fighting, they cease being people in that nation's eyes, merely a resource to be used on a whim. The Federation is not the Dominion. Jake is not a Jem'Hadar. Conscription would undermine everything the Federation is, in my opinion. "Compulsory service" is simply a euphemism for "slavery".
Had the war gone on long enough, became serious enough, I imagine Jake
would have enlisted, of his own free will, to defend his nation's freedoms- and (and this is of prime importance)
his own. If he didn't have those freedoms, why would he be willing to fight? Where's the motivation, beyond "you have to", which isn't going to be enough to keep morale strong. Why support and fight for a nation that won't recognise your freedom? One of the major reasons it worked historically was/is because young men are told they're fighting for their oh-so-helpless and adoring woman-folk at home, who, they were told, needed them to protect them. These prejudices are gone in the 24th century, though. You don't have a whole bunch of people at home who "need" you to prove yourself by protecting them, so how do you maintain morale among those conscripted? The truth of their situation would be obvious to them.
So, I think conscription would both violate everything the Federation stands for, and be ineffective at producing well-balanced soldiers. I mean, if your free society is about to be conquered, the choice is obvious; enlist to protect those freedoms, or accept conquest. If you aren't free to begin with, the only motivation is petty nationalism or because you feel you have to prove yourself in some way.
As long as there were Federation citizens breathing, I don't think you'd be short of recruits. "Personnel shortages" I think translated as "
trained personnel shortages". Getting recruits fit for starship duty, etc, was the problem, I think.