People can lead purposeful lives, if they are sufficiently fortunate. Anyone who has enough to eat, who lives in a society that tolerates a modicum of personal freedom, who has educational and employment opportunities, who is not condemned to a life of thought-destroying labor, and so on, is in a position to lead a purposeful life.
There are two and only two kinds of purposes. One kind of purpose is biological. When we say that the purpose of the eye is to see, we are talking about its biological purpose. Although philosophers have theorized about the nature of biological purposes since at least the time of Aristotle, it was Charles Darwin who discovered the conditions under which such purposes arise. Biological purposes are products of evolution. Put crudely, the biological purpose of a thing is whatever things of that kind were naturally selected to do: Eyes have the purpose of seeing because seeing enhanced the reproductive success of animals with eyes, which caused eyes to proliferate down the generations.
Biological purposes are products of evolution. So, for life to have a biological purpose, it would have to be a product of evolution. But life isn't a product of evolution. For evolution to occur, life has to be on the scene already. Evolution presupposes life, and that's why it can't be the case that life has a biological purpose.
The other kind of purpose is instrumental. When we say that hammers have the purpose of driving in nails, we are talking about their instrumental purpose. This sort of purpose comes from the intentions of a thing's creator. The purpose of a hammer is to drive in nails because that's what the hammer's creator had in mind when he or she designed it.
This position has obvious attractions for religiously-minded people. If God exists, it can't be the case that life has a purpose. A god who performs acts of creation is an agent, and to be an agent one must, minimally, be alive. So, if God exists, he must be a living god (unsurprisingly, he is described as such in the scriptures of Judaism, Christianity, and Islam). But if God is a living god, then we are confronted with the same kind of problem that we encountered when asking whether life has a biological purpose. God can't have created life, because God must already be alive in order to create anything at all. As it's conceptually impossible that God created life, it's impossible for a god to create life with a purpose in mind.
It can't be that life has a biological purpose, and it can't be that it has an instrumental purpose. Because these are the only two options, it can't be that life has a purpose.