they are very heavy when full of salt and water, so I recommend standing it on the floor, not the bottom of the cabinet. It connects to the water pipes and the drain with flexible hoses, they are usualy one or two metres long but you can get longer (they are the same fittings as washing-machine hoses, but made of food-grade materials and coloured white, not red and blue), so you can position it further away, or a plumber can extend your pipework. If you have a combi, get the higher-capacity valves and pipes (bigger diameter) for less restriction of flow.
You might have space in the corner of the kitchen, under the worktop, where it could go, but you need to access it for maintenance and to put salt in. I have a gap that can be acessed by rolling out the dishwasher; but my softener is a large model (tall) and is in the garage where the watermain comes in. If you use sacks of salt they are commonly sold in 25kg size which is very heavy to cart about and tip in. I now order 10kg bags, a dozen at a time, and I have not yet run out of the batch I ordered last June. Some machines take block salt, which is more expensive to buy.
A larger machine takes more salt and runs for longer between regeneration cycles and salt refills, which I find handy.
The space under a stone sink is quite small, and there are pipes in the way, so I doubt it will fit. They will often fit in an ordinary kitchen cabinet, though, unless you want a big one.