On the subject of prices, I suspect that the Hamburg metropolitan area unfortunately doesn't differ much here compared to Stuttgart or like with us here in Munich, just on a slightly lower level. So, if you pay the current prices, you definitely won't be the only "fool," there simply is no alternative and unfortunately you only have the choice between "staying in the rental apartment whose rent you still find reasonable" or "coping with insane real estate prices IF financially possible." The value is determined solely by supply and demand, and as it currently looks, the supply is politically deliberately not massively increased and demand does not seem to be abating anytime soon. In the sought-after big cities, inheritances still make many things possible, and even in Lower Bavarian backwaters, Polish BMW factory workers are now building one house after another (a mix of cheap land there, a lot of own work with great help from family, good shift premiums, and leveraged loans makes it possible). My parents paid around 350,000 Marks for our end terrace house at the beginning of the 80s (huge garden, 900m2 plot); recently, one of the neighbors died, and the same house type, which stood in the middle and only had a really small 100m2 garden, and honestly, was never refurbished and completely worn down, was now sold for 600,000€. "Junk," one would say at first glance, BUT: very good building substance, highway A 94 directly to Munich in 40 minutes, nice neighborhood, and very good schools nearby. The young couple in their early 30s could hardly believe their luck that through contacts to the daughter before the ad appeared on the real estate portal, they got it at this "bargain price." And even if it sounds crazy, the two are not stupid but extremely clever. The house looks old-fashioned at first glance and quite rundown inside, but if you know how beautiful the neighbors' other houses are inside, you recognize the potential, and with a little manual skill and 100,000 euros renovation money, you can make a lot out of it. While you can completely forget about a new build for 700,000€ in our area, I would not hastily rule out existing properties, depending on what is more important to you: the location, the single-family house as such, the urban-influenced neighbors, or a modern look?
Regarding the plot of land, I know many ways from acquaintances and friends: We ourselves had beginner's luck and were able to make a purchase contract for the entire plot from a planned semi-detached house (without an intermediary house provider and without being forced to build with them). A friend spent half a year five years ago in our home district 40 km from Munich digging in and letting absolutely everyone know through her parents and friends living there that she was looking for a plot and then got lucky with a private seller who heard about it. My sister-in-law did the same, involved the whole village in the house project, spoke with the mayor, networked, and was then the first to hear about the new development area. Still, it took three years overall and countless phone calls and rounds of price increases with the landlord until they can finally build there now. The total volume for the house with land (prefab house without much fuss, 500m2 land) is just under a million in a village in the Fürstenfeldbruck district; there is a kindergarten, a school, a church, and a bakery, but nothing else. No S-Bahn connection and one hour drive to work in Munich. But that is exactly what the two want: to stay in the place where they feel comfortable, real country life, peace, and not a commuter town of Munich with city people who are very different from the local rural population. A friend of my brother bought cheap building land in the sought-after neighboring town with train connection and future S-Bahn connection, took a gamble and was incredibly lucky: got the building permit for a single-family house and can now invest the full budget into the house.
However, there is also a clear counter-movement here; some of my girlfriends and their partners earn very well (Siemens, airport, BMW, consulting) and love city life and do not want to give it up. They all know that a house IN Munich with a garden without a large inheritance is simply not feasible and adapt: one bought a flat in the outskirts for a maximum of one child, another will remain a tenant in a large but affordable rental apartment with two children, and another will move within the company to another city and realize her dream of a house there.
So I would give you the tip to carefully consider what you want most as a family and what really makes you happy and then give your all for at least a year to increase your chances of finding a plot, a well-maintained existing house, a larger flat on the city outskirts with a garden share, or a building project. Think outside the box, actively ask around in your desired locations, involve people who already live there, and really look at a lot, even if at first it looks like "this is probably nothing." I didn't even want to look at our current plot at first, wasn't sure if I wanted to go that far to the outskirts and not rather want a rental apartment, and now I am so glad my boyfriend persuaded me. I'm keeping my fingers crossed for you, and most importantly: don't get discouraged and don't let yourself be talked down. It's really tough at the moment, and the place you live greatly affects quality of life, but often after a while, you get lucky or find that you can also be happy with a different path. I'm wishing you the best of luck!