A ventilator can serve three floors (one up and one down), but you should have a trained dweller work it to generate sufficient oxygen - so build the gym first, before you go down deep. The amount of oxygen available for free is less for each floor you go down, so the first four to five floors can be used safely without ventilation.
Thank you very much for your replay. I have to admit that I didn´t understand that a dweller had to operate the vent. I should have read the manual more carefully. ;-)
Now I have a nice working bunker. But I am stuck somehow as I cannot develop further or dig deeper.
Besides I found a bug. If you build storage after storage the mineral bar display will increase. If you continue building after it has reached it´s final size you will trigger the bug and the bar is broken.
Thanks again for that wonderful game.