Is buying a house for a maximum of €350,000 realistic?

Depending on the soil conditions, disposal costs, and equipment, you can expect additional costs of 60 to 100k and you have to carry loads. You are already planning with a shed. So why not put the building services on the ground floor, there you can also store cleaning supplies and other stuff; on the upper floor a small utility room for laundry and some storage space. Behind the carport a storage room for bicycles and garden tools. It is cheaper and one floor is missing. The building window must provide the space.

I would also recommend building somewhat larger and without a basement. For example, we added a 5x4m room to the kitchen, which also provides access to the parking spaces. Drinks and various outdoor items (e.g., beer garden furniture) are stored there, and one wall is set up as a workbench for my husband. We will probably put a small freezer in there soon as well. From the upper floor, we can also use the whole area as a balcony. The whole thing (without balcony flooring and railing) cost about 13k extra. Usually, you don't need the entire space of a basement, but of course you fill it up if it's there.

I would also reconsider the basement. We are building something similar, 140 sqm, and have an absolutely flat plot with very unproblematic soil. The basement alone would have cost us 70k more, in addition to the increased ancillary construction costs that come with it. It was definitely not worth it to us. We are now at 355,000 for the house alone (excluding ancillary construction costs), if that helps you further.

The basement is mainly intended to be a utility cellar (possibly a hobby room)

hit the nail on the head: using the basement as utility space is too expensive, making the house construction unrealistic for you. Prefabricated houses themselves are another 20-30,000 more expensive for a single-family house of your size than solid construction. Simply add 10 sqm on the ground floor for the utility room to the 130 sqm, with the attic in the gable roof house adding further storage space. Then a garden shed is also financially feasible. Regarding the listed prices of the houses: add 20-25% for a realistic calculation. But don’t forget the ancillary construction costs, outdoor facilities, and carport.

Mentally, we have now said goodbye to a basement and would want to try to arrange parking spaces elsewhere. What possibilities there are here... I have no idea yet

To bring some clarity to the financial options once again. We have €420,000 for house construction, incidental costs, and outdoor facilities.

Storage space where you need it.
Garden shed for garden tools, grill, seat cushions, pool for the winter
Extension carport for tools, tires, bicycles
Home technology slightly larger for cleaning supplies, beverage crates
Built-in cupboard under the stairs
Possibly a small storage room on the upper floor
Attic for Christmas decorations, suitcases, etc.

For 420,000 you get a house in the size you want ok without basement +20sqm, a carport, a garden shed, an outdoor area and the additional costs. Nothing high-end, but solidly equipped