“Let’s say that I lease an apartment for $1000/month and I intend to continue leasing it for the next 100 years and pay as I go, out of my future earnings. Does it mean that I have $1000 * 12 * 100 = $1.2 million in debt?”

No, that is not the same, unless your landlord is loaning you $1.2 million to stay in his building and you are paying him back at $1000/month with no interest. That would be a sweet deal!