a garden: a walled, watered, planted plot of ground set apart from the wild land around it
At its core גַּן names one concrete thing: a piece of ground that has been fenced off, given water, and planted. The old connection to a verb meaning to cover or protect fits what the word actually describes. A garden is bounded space, set apart from the open country, and that boundary is part of what makes it a garden rather than a field.
The same enclosed plot turns up in very different settings, and that is why this one word reaches so far. It can be an ordinary vegetable patch by a house, a king's pleasure grounds beside his palace, the divinely planted garden where the first humans lived, or, in love poetry, the body of the beloved. The senses below are not different words but the same picture used in different worlds: the everyday and royal garden, the garden of God in Eden, and the garden as a living image of abundance, renewal, and desire.
The shading depends entirely on whose garden it is and what it is being made to carry. Scholars argue less about what the word means than about how much weight a single garden image can bear, and in the Hebrew Bible it bears a great deal: kingship, the presence of God, the loss and the hope of paradise, and the joy of love.