In a sense, yes. We’re doing it right now by questioning the possibility- we both have a similar enough idea of ‘not existing’ to be able to discuss it. When you asked: “Is it possible to think about not existing,” I didn’t reply with “I like apples.” That would be a nonsensical answer- the fact that we both know that’s a nonsensical answer means that we definitely have some shared idea of ‘not existing.’
But can we actually think about, or hold a thought, which is none-existence. By definition, no. But first, lets go into what kinds of existence there are.
We both agree (I assume) that elephants exist in the ‘real’ world. So that’s one way that elephant exists. I say the word Elephant- now you’re thinking of elephants- so that’s another form of existence. Your thought of elephants is another thing which ‘exists,’ even if its only in the realm of thought. Then I say unicorn- now the thought in your mind is of unicorns- this thought, like before, is mental existence. But as you thought about unicorns you ceased thinking about elephants- your mind was focused on the concept unicorn- and the concept elephant ceased to be manifest in your consciousness. So when you were thinking about unicorns, you were at the same time ‘not-thinking’ about elephants. There was a void of thought about elephants, a non-existence. But there was simultaneously a non-existence of all things which were not the unicorn you were thinking of- a non-existence of thought about cars, love, whatever. So in a sense, our consciousness is a combination of non-existence and existence. The thing you’re thinking about at any moment exists in the mind, but that thing is immersed in nothingness.
But can you think of nothingness itself? Pure, unadulterated nothingness? Well I’m not a mystic, so I can’t say for sure. But logically speaking, no. ‘Nothingness’ could never be thought, because anything thought is something. A unicorn doesn’t exist in the real world, but the thought of a unicorn certainly exists is some sense.
To put it another way, we cannot think of nothingness because our thoughts are something. True nothingness has no existence whatever- no feelings, no concepts, no images, nothing. Our thoughts are always of one of these things. We can only think of nothingness in a negative sense- we cannot think nothingness, we can only wonder what it would be like not to think.
That being said, I believe philosophy can’t answer these contradictions, and am unsure what is truly the case.