At various times (in the United States) men have been able to opt out of war by: hiring a replacement to take their place; being a conscientious objector and taking a non-combat role; being born at the right time to not be drafted, and choosing not to volunteer; enlisting in a branch of the service unlikely to see combat; pleading a hardship; finding a doctor to claim you are medically unfit; just walking away (desertion, as long as it was not during combat, has historically be inconsistently punished).
One of the major criticisms of conscription in the US is that, in practice, whether those methods of avoiding conscription were actually available depended on race, class and how wealthy you were. So only some men could actually take advantage of them; the rest were screwed. (This conflation of the experience of a few better-off men with the experiences of men as a whole seems to be quite common in many social justice circles for some reason.)
Of the four older men whose draft stories I know personally, two had exemptions (hardship/agricultural labor), but one went to war anyway, one enlisted as a pilot in the National Guard, and one lucked out by having the draft end before his number came up. None of them were particularly wealthy or influential, although all of them were white.
At various times (in the United States) men have been able to opt out of war by: hiring a replacement to take their place; being a conscientious objector and taking a non-combat role; being born at the right time to not be drafted, and choosing not to volunteer; enlisting in a branch of the service unlikely to see combat; pleading a hardship; finding a doctor to claim you are medically unfit; just walking away (desertion, as long as it was not during combat, has historically be inconsistently punished).