The roads towards a flying house are many. More specifically, the paths towards propulsion/ levitation platform for the house are many. From zero point energy to electromagnetism to gravitational reversal, there is research being done daily to tap the endless energy flux of the universal multidimensional cloth into which we are all fully embedded.
By the way, nobody knows how gravity works. We see it in action and can describe it well enough to land rovers on Mars but the underlining principles of gravity are not nearly as well understood as electromagnetism or the two types of basic nuclear attraction. There are unsuccessful attempts to extend the quantum theory to gravity and there is speculation about these "graviton" particles but so far that theory appears very ephemeral and weaker than most.
For the hell of it, lets throw in a part of the final paragraph from Nietzsche's final unfinished book about the nature of the universe.
"And do you know what "the world" is to me? Shall I show it to you in my mirror? This world: a monster of energy, without beginning, without end; a firm, iron magnitude of force that does not grow bigger or smaller, that does not expend itself but only transforms itself; as a whole, of unalterable size, a household without expenses or losses, but likewise without increase or income; enclosed by "nothingness" as by a boundary; not something blurry or wasted, not something endlessly extended, but set in a definite space as a definite force, and not a sphere that might be "empty" here or there, but rather as force throughout, as a play of forces and waves of forces, at the same time one and many, increasing here and at the same time decreasing there; a sea of forces flowing and rushing together, eternally changing, eternally flooding back, with tremendous years of recurrence, with an ebb and a flood of its forms; out of the simplest forms striving toward the most complex, out of the stillest, most rigid, coldest forms toward the hottest, most turbulent, most self-contradictory, and then again returning home..."
On that note, today we start looking at electromagnetism. The video below shows that it's no joke.
What needs to be done?
1) Tapping the permanent magnet of planet earth and locking the flying house into computer guided locked trajectories. Similar to how liquid nitrogen is able to capture magnetic fields in the video below:
Tom Shannon is a cutting edge magnetic sculpture artist. In the video below at minute 7:00 he shows the sheer extent of what is theoretically possible. Yes, it is a floating island but the principle is much the same as the locked superconductor disc shown in the video above. Only rather than 500 micron Sapphire crystal, the bottom of the island is a layer of diamond (once nanotechnological assembly later in this century makes diamond production even cheaper than glass currently).
At 8:45, we get to something smaller and closer to what we want with the Air Genie. It is a spherical flying blimp that surfs the magnetic field of the planet just as the tiny robotic magnetic sculptures do within Shannon's studio. Air Genie shows a combination of magnetic levitation with the tensairity floating theory of Buckminster Fuller.
A good analogy is the series of Tokamak fusion reactors that showed that fusion is possible even if the input of energy was greater than the output (thus not accomplishing the main goal yet paving way to projects such as ITER).
This would demonstrate various problems and inspire second generation attempts among pioneers in the future.
These pioneers will most likely be transhumanists from rich families inspired by the daredevil playboy aristocrats like Howard Hughes a century earlier. They'll want to recreate and experience the same thrills and fame that their tycoon great grandpappy experienced. Partying and environmental/communist activism doesn't provide nearly the same kicks.
A good way to influence them is to emphasize that by the middle of 21st century, a large swath of humanity may be merging together into a "neural internet" aided transhumanist collectivist entity. Now it may very well be that it will be lots of fun to be part of that techno-fleshy carpet on the ground. However, the hyperindividualist playboys may want to keep their options open just in case and construct highly mobile, cloaked, and comfortable "air yacht" shelters up in the air to be safer from the madness below on the ground (we're not even mentioning the possibility of global decades long "civil war" between haves and have nots).
Another way to influence the youth is to popularize and constantly innovate magnetic toys (preferably Internet connected and controlled by brain waves). This will get little Richard Branson clones away from the increasingly amazing virtual reality games somewhat and towards actual external physical world. In turn, the desire to have things float and direct it with their minds will translate into air vehicles and eventually flying houses. Human psychology works in such a way that it is always best to suggest the craziest maximum possible idea (think utopian ideologies in 19th century) and then have technology and society meet it halfway eventually.