For most of us, the answer isn't a separate building, a closet, or even the garage. The single most popular alternative is right under your nose (and the rest of you, too, for that matter) in the cellar below.
Frequently, the space downstairs is unfinished. If the furnace is there, it's heated. Count two advantages already.
The separation from the rest of the house is hardly total, but there's at least some soundproofing and a certain amount of the dust and dirt will be contained.
If there's a concrete floor, it's a solid, practical base for even substantial stationary power tools. A well-equipped shop needs a structure able to support heavy workbenches, fixed saws, and the other stationary tools. The floor surface must also be durable and not so precious that every scratch and mark will make you wince. And it probably needs to be resistant to solvents and grease.
It's true, cement floors are hard and cold, but otherwise they are just about perfect. You don't like the grayish color? The dust is hard to control?
Apply a concrete paint to seal the surface of the concrete. It'll cut down on dust and brighten the space, too, as paints are available in light shades and colors.
All of that is good news. On the other hand, dampness is a common problem in cellars. Make sure there is plenty of ventilation in yours if you plan to put your shop there, because moisture can ruin tools almost overnight. Condensation and minor dampness can often be addressed with a fan and a dehumidifier, but if the place frequently floods or has standing water in it, you will probably want to keep on looking for your workshop space.
What about access? Can you get a sheet of plywood down the stairs? You don't want to set yourself up to make the mistake of building a captive project, like a boat or a bureau that can never escape the basement. It happens.
What about other users of the space? Is there already a play-room for kids in your cellar, or a utility space with a washer and dryer? Noise, safety, and dust can become important matters when the other inhabitants might be inconvenienced by the arrival of a workshop. One common (and relatively simple) solution is a partition, but it isn't the magic answer in all cases.