You certainly can have evidence against something that doesn't exist. There is very strong evidence that there are no (ordinary) elephants in the room I'm in right now, for instance: elephants are large and easy to see, and the fact that I see none when I look around is evidence for their absence. Direct evidence that there are no invisible miniature elephantoids made from weakly-interacting Dark Matter in the room, of course, is harder to come by.
Similarly: If someone believes in a god who regularly answers prayers, works miracles, etc., then a shortage of miracles and surprisingly answered prayers is evidence against the existence of that person's god. If someone else believes in a god whose primary objective is to avoid notice, it's hard to see how anything could be evidence for or against its existence. (And, by the same token, it's hard to see why anyone would care much whether it exists or not. A god whose existence we can't get evidence about is ipso facto one that makes little difference to how the world is.)
... Except that it's pretty plausible that some sort of formalized Ockham's Razor works, where your initial evaluation of how likely something is looks more or less like 2^-(length of shortest complete description). In that case, a universe with a maximally-self-hiding god is much less likely than a universe without one.
Anyway, most theists believe in gods that are (perhaps intermittently) active in the world: answering prayer, healing sick people, raising people from the dead, etc. There's no reason why there shouldn't be evidence for or against the existence of such a god. (I think the evidence is heavily against; others disagree.)
There is very strong evidence that there are no (ordinary) elephants in the room
That's assuming a closed system open to inspection.
The universe is neither of those. I can say what is or isn't in the room because I have total control over and full freedom to inspect the room to the level of detail required to find elephants. Your example simply doesn't translate.
You pivot then into the "far away God is the same as missing God" argument. That's another can of worms (One which I did not go into)
What I find humorous is that every 80 years or so we see a resurgence in either atheism or deism. Each time the generation feels like it's way smarter than the one before it. Only the arguments are all the same as they were three iterations ago -- or three thousand years ago for that matter.
The existence and nature of a God, if there is one/them/it, isn't as easy as looking for elephants. But that's what makes the conversation fun -- as long as everyone acts in a civil manner.
<i>Your example simply doesn't translate</i>: I wasn't saying "God is just like an elephant", I was saying "Your alleged point of logic, claiming that one cannot have evidence that something doesn't exist, is incorrect". The elephant is a counterexample to it.
For what it's worth, though, I think the argument does translate -- for some but not all notions of God. Specifically, as I said before, if you adopt a notion of God that involves not-impossibly-vague claims about how he interacts with the world we can observe, and if those claims are something other than "he basically doesn't", then it is possible for there to be evidence for or against the existence of such a God.
This is true even without "total control over and full freedom to inspect" the universe. (Just as the claim "Some star within 100 light years of us went supernova between 100 and 200 years ago" is one we can have evidence for or against even without total control of and full freedom to inspect everything within 100 light years of us.)
<i>You pivot then ...</i>: What I'm saying is not "far away God is the same as missing God" but "professedly inactive, or indiscernibly active, god is hard to distinguish from no god". It seems to me that there is a difference.
<i>What I find humorous is ...</i>: You're welcome to find humour wherever you please, but for what it's worth I don't think I'm way smarter than people in the generation before mine, and I don't think skeptics generally think they are. (I expect skeptics generally think they're smarter than average people in the previous generation, but no more than they think they're smarter than average people in the present generation. Whether they are is a separate debate.)
Similarly: If someone believes in a god who regularly answers prayers, works miracles, etc., then a shortage of miracles and surprisingly answered prayers is evidence against the existence of that person's god. If someone else believes in a god whose primary objective is to avoid notice, it's hard to see how anything could be evidence for or against its existence. (And, by the same token, it's hard to see why anyone would care much whether it exists or not. A god whose existence we can't get evidence about is ipso facto one that makes little difference to how the world is.)
... Except that it's pretty plausible that some sort of formalized Ockham's Razor works, where your initial evaluation of how likely something is looks more or less like 2^-(length of shortest complete description). In that case, a universe with a maximally-self-hiding god is much less likely than a universe without one.
Anyway, most theists believe in gods that are (perhaps intermittently) active in the world: answering prayer, healing sick people, raising people from the dead, etc. There's no reason why there shouldn't be evidence for or against the existence of such a god. (I think the evidence is heavily against; others disagree.)