How much energy do you need to levitate/counteract gravity without any displacement or change in height?

A table can forever keep an apple "levitated" above the ground with it's normal force. That requires no energy. No work is done.

A force does not spend energy to fight against another force.

The force may cost energy to be produced, though. This is a separate issue. The jetpack spends fuel to produce an up-drift force, the human body spends nutrition to extend/contract muscles to produce the "holding"-force to hold a milk can, but the table spends nothing to produce it's normal force.

The jetpack falls down after a while and you feel tired after a while, not because work was done on the objects, but because work was done inside those "machines" (jetpack and body) that produce the forces. The table never gets tired. It never spends any work.

The issue is clearly not about holding anything. It takes no energy to hold stuff. You are correct that no work is done on the levitating man, if he undergoes no displacement. Work may be done inside the "machine" that produces the force, but that is internal.

For more, here's an article on the human body "machine" about this very topic.

To hover in the air conservation of momentum dictates that to keep a 100 kg object hovering we must "throw" 100 kg down towards the earth at a velocity of 9.8 m/s for every second we want to hover. The kinetic energy required to accelerate 100 kg to 9.8 m/s is 4.8 kilojoules. So a propeller that grabs 100 kg of air per second would need 4.8 kilojoules per second or 4.8 kilowatts (a watt is joules per second).

We could also propel twice the mass of air at half the speed to hover. Since kinetic energy is the square of the velocity propelling 200 kg down at 4.9 m/s would use 2.4 kilowatts or half the energy. So bigger is better and there is no theoretical limit to how low your energy consumption can go. Some sort of futurist tractor beam that can push or pull a very large mass of air would use almost no power. With our current technology and materials a very large open blade (i.e. a helicopter), large ducted fan or high-bypass turbofan are your best options as they will move the maximum amount of air with the minimum amount of energy.

If you want a more traditional jet pack where all of the reaction mass is kept on board then you're getting into rocketry and don't care about energy efficiency. You just care about the specific impulse (energy density) of the fuel. Also, even with the best rocket fuels your maximum flight times will be measured in seconds.

Your question is actually profound in a subtle way. The key to understanding this is that the man has a force being applied to him by gravity that is pulling him down. In order for him to stay aloft at a constant height, there must be a force that acts in the opposite direction and counteracts the force of gravity.

In your example, that counteracting force is supplied by the jetpack. So the jetpack must continually produce an acceleration upwards of equivalent to the weight of the man (and the jetpack.) But why is that different than when the man is standing on the ground? The earth's gravity is still acting on you but you don't have to continually burn fuel to stay in place. Consider the same standing on a large spring. When he first get's on it, he will move towards the earth and compress the spring until it pushes back enough to stop his motion. The spring's upward force is being supplied by weight of the man in a reflective manner. Essentially the ground does the same thing. The elasticity of the surface creates a mechanical equilibrium. Newtonian models don't really describe how materials produce elastic force. It is simply assumed.