Paul's instruction sounds impossible until you notice he gives it a mechanism. Be careful for nothing — that is, anxious about nothing — is not a command to try harder at calm. It is followed immediately by but in every thing by prayer and supplication with thanksgiving let your requests be made known unto God. The anxiety is not suppressed. It is relocated.
Notice in every thing. Not the large and worthy concerns only. The small nagging ones, the ones you would be embarrassed to say aloud, the ones you suspect are your own fault. Everything is admissible. Nothing has to be tidied up before it can be handed over.
The promise is not that you will get what you asked for, or even that you will understand what is happening. It is that the peace of God, which passeth all understanding, shall keep your hearts and minds. The word translated keep is a military one: it means to garrison, to post a guard. Peace is described here not as a feeling that arrives but as a sentry that stands watch over the part of you where the worry gets in.
That is why the peace can pass understanding. It is not conditional on the situation making sense. It is stationed at the door regardless.