Depends on how "soft" you are making the sci-fi.
The bottom of the carpet is lined with (as yet undiscovered in real life) particles which repel each other almost like quantum springs when exposed to... I don't know, oxygen? Before the flight, the carpet needs to be placed into an extremely powerful compressor which squeezes them together and stores up the energy. Then the repulsive action causes thousands of individual little "jumps" between the particles, which forces the carpet to rise from the ground.
Here's the hard part: Control. Using a complex system of threads weaved within the carpet, the shape underneath can be changed, making subtle little bends and folds which can increase and reduce the repulsive force in different sections by activating more or less of the particles. In the hands of someone of great skill, this technique can actually be used to direct and steer different parts of the carpet.
Now, about flying throughout the "Seven Seas" (i.e. the entire world), it could be that a number of linings of the stuff would be needed, so pilot might need to have a way of removing a layer of it, mid flight.
You'd also have to have a special airtight cover to land on, so that when you got off, the carpet didn't fly off uncontrolled.