The famous atheist philosopher and mathematician Bertrand Russell was once asked to suppose that he'd died and found himself face to face with God who asked him to account for his lack of belief. What, Russell was asked, would he say? Russell's reply was a curt, "Not enough evidence."
This has been a common response to similar questions for centuries. The unbeliever argues that the burden of proof is on the believer to demonstrate that God does exist. Failing that, the rational course is to suspend belief.
In the lapidary words of 19th century writer William Clifford, "It is always wrong, everywhere and for anyone to believe anything on insufficient evidence." Of course, Clifford would presumably want to exempt his own statement for which there's no evidence whatsoever.
In any case, a claim for which their was no conceivable empirical test was considered meaningless since there was no way to ascertain its truth or falsity. This evidentialism or verificationism, as it was called, enjoyed considerable popularity back in the 1930s and 40s among those who wanted to make the deliverances of science the touchstone for meaningfulness, but it eventually fell into disfavor among both philosophers and scientists because, rigorously applied, it excluded a lot of what scientists wanted to believe were meaningful claims (for example, the claim that life originated through purely physical processes with no intelligent input from a Divine mind).
But set the verificationist view aside. Is there, in fact, a paucity of evidence for the existence of God or at least a being very much like God? It hardly seems so. Philosopher William Lane Craig has debated atheists all around the globe using four or five arguments that have proven to be exceedingly difficult for his opponents to refute. Philosopher Alvin Plantinga expands the menu to a couple dozen good arguments for theism.
So how is this plenitude of evidence greeted by non-believers? Some take refuge in the claim that none of these is proof that God exists, and until there's proof the atheist is within his epistemic rights to withhold belief, but this response is so much octopus ink. The demand for proof is misplaced. Our beliefs are not based on proof in the sense of apodictic certainty. If they were there'd be precious little we'd believe about anything. They're based rather on an intuition of probability. The more probable it is that an assertion is true the more firmly we tend to believe it. Indeed, it's rational to believe what is more likely to be true than what is less likely.
Could it be more likely, though, that God doesn't exist? There really is only one argument that can be adduced in support of this anti-theistic position, and though it's psychologically strong it's philosophically inconclusive. This is the argument based on the amount of suffering in the world. When one is in the throes of grief one is often vulnerable to skepticism about the existence of a good God, but when emotions are set aside and the logic of the argument analyzed objectively, the argument falters (see here and here for a discussion).
This is not to say that the argument is without merit, only that it doesn't have as much power to compel assent as it may appear prima facie to possess. Moreover, the argument from suffering (or evil) can only justify an atheistic conclusion if, on balance, it outweighs in probability all the other arguments that support theism, but this is a pretty difficult task for an inconclusive argument to accomplish.
Actually, it seems likely that at least some who reject the theistic arguments do so because they simply don't want to believe that God exists, and nothing, no matter how dispositive, will persuade them otherwise. Even if God were to appear to them, a phenomenon some skeptics say they'd accept as proof, they could, and probably would, still write the prodigy off as an hallucination, a conjuring trick, or the consequence of a bad digestion. In other words, it's hard to imagine what evidence would convince someone who simply doesn't want to believe. Jesus himself once said precisely this (see Luke 16:19-31).
I'm reminded of something the mathematician and physicist Blaise Pascal said some three hundred and fifty years ago. He was talking about religion, but what he said about religion is probably just as germane to the existence of God. He wrote in his Pensees that, "Men despise religion; they hate it and fear it is true."
The "not enough evidence" demurral is in some instances, perhaps, a polite way of manifesting the sentiment Pascal identified.