How much financing? Realistic budget?

Well, that’s unnecessary since the property is not built on.

The 2500€ includes allowances, but I also have to say that I only finished training last year. Currently I work 39 hours a week, after parental leave I would like to work again at least 50%, preferably night shifts, that shouldn’t be a problem at my workplace.

You calculate:
Land plus incidental purchase costs
2500€/sqm
30000€ outdoor facilities.
Additional construction costs 50000€
Basement 80000€
I am also at 550000€ including land
but you can also build with a basement/living area instead of a basement.
Then the 80000€ is omitted.
All just rough estimates.

But I wonder how you imagine this: man self-employed and gets no loan, you at most 300000€, parents should already stand in again for your young life, 45 minutes of travel time each for everyone, (gasoline and car costs are rising), during parental leave one salary is lost, later you have to spontaneously pick up a child from daycare, etc.
I don’t see all of that.

You can do it: you work at night, your husband during the day. Everyone watches the children separately. In between, the daily work is passed on to the other on a note and the garden is neglected anyway.
Everyone lives their life - at some point no longer together.

Honestly, I think you should still wait. Savings rate and what is left monthly, I think was different?
So there would still be more room to save.
Then think in the medium term about where the living point should be: 45 minutes travel time for both is bad. There should be Plan B and C.
When Corona is over, the self-employment will also be established so that the loan will be higher.
But not like this.

With a salary under 4500€ almost nothing works. Unless equity or EL is relatively high. All circumstances like commuting costs, also the parental leave in the future speak against the wish.
Most here save 10-15 years and have a salary of about 3500€, the woman then 1800€ part-time. Then 150000€ equity is not uncommon.
With your rent and additional costs I wouldn’t handle 1500 plus 400 additional costs anyway - I would use that as long as possible, building up a savings cushion of a six-figure amount. With this rent you can also bridge parental leave well.

I still say: save until you are in your early 30s, then you have a lot of equity and then it works. The question is, how urgently do you want to pursue the desire to have children?

Maybe you could realize the desire to have children first and at the same time stay as cheaply as possible with the parents? When the two children are there and you both work again, the financial situation (apart from the crazy commuting) is stable and plannable. And the children are out of the worst. The goal could then be the own home by the first child’s school enrollment. Nevertheless, it stands and falls with the discipline of saving.