I have many small sabilas (a cactus), so my idea was to cut the top part of several wine bottles, fill the remaining glass container with water, and place the cactus in the bottle, like in the picture below.
My question: can a cactus live in water or will it eventually die?
In short, no. It can't live with its stem submerged in water, it will die, and the one you show in the photograph is clearly very unwell and probably on its way out, showing some evidence of rot.
It is possible to grow cacti hydroponically - this is said to be easier than growing in soil, though when you read the list of instructions, which include removing the cactus from its substrate every fortnight and washing the roots, I'm at a loss to understand why anyone would think it was easier, but perhaps some people might find it so. Either way, you need aggregate, a mesh inner pot and an ordinary outer pot, and the roots of the plant should be anchored firmly in the aggregate, with water never allowed to be in contact with the stem of the plant. Further information on the hydroponic method here https://www.hunker.com/13426578/how-to-grow-cactus-hydroponically
