I was wondering if it's possible to do the same without drugs somehow?
I believe so, but I can't guarantee that others have the same experience.
In my experience the long lasting anti-depressive effect of psychedelics isn't all that different from other intense experiences. Vacations have had a similar effect for me.
Psychadelics are a good reminder that it's possible to feel and see things differently. It relativizes thoughts and emotions, in a way which is hard to explain.
Compared to a vacation or other environmental changes, psychedelics doesn't require much of the user. They're also cheap and time efficient. A four hour trip can do as much for the mind as a full week vacation.
They are however unpredictable, illegal, and sometimes dangerous. When things go wrong you can experience a week of pure hell in just a few hours.
If you're able to, try a vacation to an unfamiliar place. To me that's the essence of what a psychedelic trip is.