Where to put the fitness room?

That's what happens when you want to build without a basement. With a basement, you wouldn't have that problem. An extra room outside the house means you lose garden space. Whether this is relevant for you or not, you have to decide. Furthermore, such a room costs money not only for construction but also for maintenance, meaning heating in winter. And how accessible is the room? I mean, do you have to leave the house and first go through the garden to reach the room? That probably wouldn't be very pleasant in rain, ice, or snow. So either reconsider a basement and try to minimize the additional costs through appropriate personal effort or provide a multi-purpose room.

A garage is typically unheated. That might not be a problem, I just wanted to mention it. The cheapest additional space can be created under the roof. If possible: steeper roof and suitable roof truss and then use the attic space.

Because basements are built for free... but well, wrong discussion.

If you really want this room and not just because everyone has one, you have to prioritize... e.g. build 15 sqm bigger and only go on vacation once a year - or have a basement and no vacation at all :D
If you only want to train now and then, no one will say anything against the weight bench in the unheated garage without windows.

This statement

Im Haus selber geht wohl einfach zu viel wohnfläche kaputt.

is somewhat general. An interesting detail would certainly be a floor plan. Often it seems that a lot of space is simply "wasted" that could have been used sensibly.

A fitness room in the basement also has a disadvantage. The ceiling height. Bumping, tearing, and other overhead exercises would require about 2.75m of room height at my height.

Gym is nice and all. For those who have the time.. (travel, waiting times during peak hours etc.)

Good morning!

Does such a room count as living space in the "construction-technical sense," so that, for example, I have to comply with the setback distances?
Is a garage only a garage if there is an actual possibility to drive a car into it?

What I am getting at: building a garage directly on the boundary, without the possibility of access by car and using it otherwise (e.g., as a sports area) will probably not be feasible, right?

Regards,
Zehn0813