Yes and no. It can mean "to ask for help," but it doesn't really mean "ask," it means something in between "ask" and "command."
To "enlist" is not quite a joke, but it is somewhat figurative language. It carries the idea of someone in a position of authority, such as in the military, enrolling people or putting them on a list... assigning a group of people to a task. It carries an idea of commanding or ordering people to do something.
In everyday life, it applies to situations where someone can't really give commands, but is trying to urge people to take on a task. There are a number of similar expressions. Let's say there's been a group picnic and there is a certain amount of trash or litter on the ground. The leader might say,
"Before everybody leaves, I'd like to enlist some help in picking up the trash," or
"I'd like to corral a few people to help pick up the trash," or
"I need to grab a few people to help pick up the trash."