Since you referred to intuition, this is possibly not too off-topic.
In a sense, the archeotipe archetype of the topologic categories is, how very elementary beings perceive the world. If I was an amoeba, I'd possibly just understand space as places close, or less close to me, not otherwise structured. I'd have no particular metric idea of my own shape; I'd just feel more or less connected, &c. So, a possible answer to your question is: like a dull amoeba.
To make an example possibly closer to us, think you're in a car in the urban traffic. Due to one-way streets, metric is not the best way to organize your perception of the space: actually, the proper topology to do that is possibly not Hausdorff (usually, you can't move to A without immediately finding yourself in B, and once you are in B, you are enormously far from A, even if you changed your mind about the opportunity of the movement.)