Can we afford the financing?

Okay, I overlooked the weekly shopping ;-)
Do you have an energy certificate for the house? How extensively was the house renovated? It can happen that the price doesn't stop at the purchase price of the house. In that case, 80 € for gas might also be too little. Let's assume nothing was done and the house consumes 400 kWh/m²a, that would rather be 190 € / month (but that is somewhat extreme).
You could probably still save on dental insurance if money gets tight - that is usually not a real insurance benefit - you need implants depending on your teeth rather when you are 60 or 70. A savings plan with 40 € per month over 30 years might be better there.
So you don't need to have any worries... the rate is reasonable - your expenses are a bit messy :D

Hello BeHaElja,

at the moment, there is an oil heating system in the house that needs to be replaced (new law, as it is older than 30 years). A new oil heating system + renovation will be done here, or if the budget allows, a complete conversion to gas (this buffer is already included in the financing and is expected to be co-financed cheaply by Kfw.
The house has a fireplace in the living room. Hot water runs through an instantaneous water heater.

Yes, an energy certificate is available:
Energy demand 165.6 KWh (m²a) (of which 143 oil + 21 electricity)

Overall energy efficiency:
217.1 KWh (m²a)

The sellers also said that they mostly only heat with the fireplace and only use the oil heating system in harsh winter, they said something like 400 liters of oil per year.

Renovations included a new roof (but not fully insulated) done a few years ago.
Otherwise, new radiators inside and plastic insulated windows + door.
As mentioned, it is a "mid-terrace house," which means that on both sides it is already well insulated by the neighboring houses.

The seller wants to sell the house... no one heats with a fireplace in winter -.- If the fireplace is not that great, you might not even want to use it because everything always smells. 220 kWh / m²a would thus, without further renovations, be more like 110 € than 80 € (gas) - but still within range. 220 kWh/m²a is 220 kWh/m²a even if the house is 30 m underground and a bunch of penguins are waddling on it ;) What does not fully insulated mean? Maybe you have to insulate the top floor ceiling (by law) - but with a terraced middle house and the consumption, that certainly makes sense. Is the attic developed?

Yes, well that’s true. I am familiar with sales tactics. Yes, the attic is converted. I can only say that I was in the attic at 35° outside temperature and it was comfortable. So it can’t be that poorly insulated. For comparison, in my parents’ house, I feel like it’s 40° in the attic when it gets a bit warm outside :-)

Yes, a few euros back and forth, you don’t heat all year round anyway. It’s calculated upstairs for every month. In addition, you have to consider that there is an ancient oil heating system inside. A new, modern one already consumes much less.

Mainly, it’s that I’m getting stomach aches and I really don’t know whether I should take this step or not. I looked for rental apartments here in the Stuttgart area (30km radius), under 800€ cold rent, you really can’t get anything decent.

The energy performance certificate considers the entire year .... 220 kWh/m²a * 100 m² * 1 year = 22,000 kWh - if you get that from a gas supplier, it amounts to €110 ... and that every month, not just in winter.
With your second point (new heating) you are of course right - but better calculate with the worst case first and be happy if you can still go out to eat.
A quick question - how long is your fixed interest period and how high is your repayment?

We are also quite young (30 and 27) and are building now too... many also advised us against it - ironically all those who built at the same age and also with children ;) If you have calculated everything well and sufficiently for yourselves, you plan to stay together for the foreseeable future, and you believe it is the right thing for you, take the step.
If you feel too uncertain, you can also wait 2-3 more years and save during that time - maybe that will then give you the necessary security.

The monthly energy costs must be clarified: If there is no energy certificate, then the invoices from the previous owners of the last 3 years must be reviewed. Is there really no need for renovation in the old house? Before purchasing, the property must be inspected by a building expert (350-400 €). The expenses for car comprehensive insurance seem very high to me. Possibly switch to an online insurance!