Separate the basement room or do it yourself?

The costs according to @nordanney are generous. In the basement, any door from the hardware store will do.

Yep. But with the current prices and since I don't know what the OP's demands are, better a bit generous.

Thank you for your answers

when you walk down the stairs, the technical room would be directly to the left and along the corridor to the right about 3-4m the storage room

from the stairs you would walk straight towards the new drywall, where the new door would have to be installed

we have underfloor heating in the basement would you have a separate heating circuit made for the storage room and the corridor each (additional cost) if you make corridor and storage room into two rooms? it would also need its own light switch

the drywall would only be noticeable later if you knock on it, it would probably sound hollow compared to the sand-lime brick wall, right?

To be honest, I don't understand why you want to separate the anteroom at all? Did I understand correctly that there is a storage room behind it anyway? And you want to separate the hallway to make yet another storage room out of it?!

My recommendation in descending order of priority:
1. Just leave it as is. Eventually, you'll regret it because you want to do something different after all.
2. Suggest the drywall variant to the general contractor, maybe it will be cheaper then (no plaster needed, just spackled).
3. Do it yourself.

in the basement we have 3 rooms

technical room storage room and anteroom

technical and storage room are 2 rooms closed off with doors and the anteroom is simply an open room when you come down the stairs

we want to make a storage room and hallway out of the anteroom.

the new small storage room should be closed off with a wall and door so it doesn't always have to be tidy and you are not immediately in the storage room when you come down the stairs

I quickly made a sketch of the floor plan

the red would be the new wall, the P arrow the stairs, there would be a small hallway to access all rooms

On the left [technik] and on the right should be the fitness room and the middle one a storage room

Looks like a Saturday of work. Screw the profiles on the right, top left, and bottom (including matching profiles for the door). Slap the boards on and putty/sand multiple times.