Thanks already for so many answers!
I already suspected that it will be really tight. But a basement is unfortunately out of the question price-wise. A small attic is available, but more suitable only for things like suitcases, Christmas decorations, old files, etc.
Maybe we really have to think about a shed at the double carport. It will be tight with our property boundaries then.
In Lower Saxony, we have to keep a distance of 3m from the property boundary with our house. Border development is allowed on one side up to 9m, but a maximum of 15m in total. We still have to find out exactly what actually counts as border development (if the carport, for example, starts 2m from the boundary, is it still considered border development?...)