Does the ARCA hoverboard violate known limits on small-diameter electric fans?
The key is that the Arca board only works in ground effect. Therefore, the pressure difference over the area of the board is the key for calculating its power needs.

From the video I would estimate the size to be 0.6 x 1.2 m thanks to @mins more thorough research we know that the board measures 1.45 x 0.76 m for an area of 1.1 m². To lift both the occupant and the board itself it claims to create 430 pounds of force, equivalent to 195 kg of mass in Earth's gravity, so the pressure increase between top and bottom is 1740 N/m² or just 1.7% of atmospheric pressure. This is not much when you consider that a turbofan compressor stage can create a pressure differential of 1.7, albeit at much higher speeds.

The concept is basically the same as the Hiller flying platforms of the 1950s, the first of which was powered by two 44 HP engines. Since the intake geometry of the Arca board is - how should I say - less than optimum, it can not fly as high as the Hiller vehicles and I find the claimed 272 HP hard to believe and the high power draw is due to the low efficiency of 36 closely coupled blowers.



Hiller VZ-1 Pawnee (picture source)

To calculate the exact amount of power needed I would need to calculate the mass flow from below the board at the given pressure, which depends nonlinearly on altitude. Then I would need to calculate the power to produce this mass flow at the known pressure differential with 36 small blowers. Sorry, but even my Hoerner books do not cover this. What is certain is that the thrust from the 36 blowers is not nearly sufficient to lift the 195 kg when out of ground effect. The board only works at very little altitude when the outflow cross section around the board is restricted. This gives it an automatic altitude limit of a few centimeters and makes it self-stabilizing in pitch and yaw.

It takes some thick skin to watch the full length of this platitude-filled piece of marketing for an obvious gimmick. The camera angles were chosen to make it look fly higher than it actually can, and in the one shot where it does fly higher you cannot see who or what is standing on top. Anybody spending 20 grand on this is going to regret it quite soon.



EDIT:

With the answer provided by Bruce Abbott to @mins' question on Electrical Engineering SE the question can be solved. Bruce approaches the problem from the electrical side and links to a 120&nbsp;mm blower which creates 74 N thrust and consumes 7 kW of power. He calculates 65 kg for the batteries and 36 kg for the blowers, for an estimated mass of the board of 110 kg. However, in combination the thrust from closely coupled blowers will certainly be lower than that of an isolated blower with ideal flow conditions.

Conclusion: Looks possible, no limits are busted, but there is some potential for further optimization and badly needed mass savings on the board.