Is house construction financing possible at all?

The thing about getting younger is just a phrase.
My parents built at 60/62. We are currently building at 41/43.

You just have to be able to afford it at that age. If you build at 60 and bring equity from the sale of a previous property, no problem. If you can manage with a low installment, also no problem. But let's be realistic. We are currently talking about nearly half of the net income as a pension and that still has to be taxed. In retirement, you have plenty of time (because you’re no longer working 40 hours) and might want to do things as long as you’re physically fit. You can afford a significantly lower installment than during your working life.
And don’t forget... in 20-30 years the houses won’t be new anymore but will gradually require some investment.

What kind of job brings in a meager 1000 euros per month full-time? I would generally try to improve the woman's income situation. Maybe use parental leave for further training instead of being stuck for life in the < 800 Euro segment. With the pension prospects, the house is of no use if you cannot additionally provide privately beyond that.

I personally consider your building wish unrealistic at the moment. Stay a little longer in your rental apartment, continue to build equity, and wait.

In the inventory, you can either keep your ears open and buy a house before the seller's first conversation with a broker, or go house hunting with a fixed affordable limit. The limit has already been discussed with a bank and is financeable. This way, you simply enter negotiations with the seller. You already have a "no," and the seller can't find you any more than audacious.

What needs to happen so that we can afford it? Significantly higher equity?

Yes, save a lot now until an affordable existing house comes along.

How much and/or installment payments high (thus higher salary or new job? :) ).

Yes, first your wife should go back to work. And yes, if possible also a better job. €1000 for a full-time position unfortunately sounds like minimum wage :(

My tip: A) Don’t build - build equity B) You live in the countryside, keep your eyes open, ask around - I see it here in our area - 50% of the houses are sold privately. Often on absolutely fair terms. C) Never let your wife work full-time for 1000 euros net again. I know many young mothers who, for example, came from completely different fields and now work at Aldi for 16-18 euros an hour. Just the extra 300-400 euros per month can already be crucial for a proper financing and more flexibility for you.