Mrthfx

In a world with practically unlimited energy and advanced technology they have no need to burn fuels for energy or even use fire. To travel around their huge planet cities they use aircraft and hover cars that use dense compressed air to hover and propel themselves. They use a practically indestructible material to hold xenon or other heavy gases at very high pressures.

I know that they would likely be noisy but I am unsure of the scale of how much air a car would need to carry to fly for days before refueling. Would the weight of all the air make it too heavy to even take off or move at a reasonable speed?

"Yes", but "No".

For a short ranged hopper - Getting from A to B locally where you park the vehicle in a top up station before going about your day - compressed gas could work. If the trips are short enough. If they're flying, they'll be VERY short trips. [Poking around on Google can come up with light cars/bikes running on compressed air with a range of a few miles.]

However there is a limit to how far you could push such a technology, and you quickly run into several problems.

Eventually you hit a limit on compressed Gas, and will have to reach to liquid storage to fit any more into a tank. Thanks to the ideal gas law, and common phase-change physics, this then jumps the energy required to actually use it - As you try to use the physical energy stored in the compressed gas, you need more thermal energy exchange to let the gas keep expanding.

As you attempt to expand your range, you are hit with the problem of fuel-tyranny: To carry fuel/energy, you need to burn/use fuel/energy to get it to the place where you will use it to move farther.

Say you have something that uses X fuel to move Y distance. At first glance it is easy to assume that 2X fuel will give you 2Y distance, but you need to use fuel to carry the extra fuel... So you add more than 2X, but might need to add more power/thrust to actually move that much fuel, which in turn means you need more fuel to provide it, and... Well you can see how that quickly starts to run away. [If you don't see that, go play Kerbal Space Program, and 'add more boosters']

Beyond that there is also safety issues with compressed gas. "Heavy" gasses can displace normal atmosphere, and come with smothering risks. Even just regular gases come with the risks of critical failures that can make them more risky than traditional fuels past a specific energy density.
Compare the risks of a 'small leak' in a tank of jet fuel - It slowly leaks out over time. Even if it is on fire, that energy is dispersed steadily over time. If you rupture a compressed gas tank, then the nature is that it will want to expend nearly all of its energy in a very short time. [And gets extra interesting if the compressed gas is reactive, as it will want to violently force itself out of storage from even a relatively small failure.]

In short, no, this would not be a practical vehicle. Certainly it wouldn't provide you with days of endurance.

Your vehicle is basically a less efficient (albeit safer) rocket. Rockets use combustion to increase the pressure and temperature of their exhaust as well as expel it, but after that the principle is the same: stuff leaves your vehicle in one direction, your vehicle gets pushed in the opposite. However, this means that you run into perhaps the biggest problem in rocketry. Your thrust has to carry not just your vehicle's frame and its payload but all of its unused fuel (or unreleased pressurized gas, in this case).

This leads to the Tsiolkovsky rocket equation, one of if not the most famous equations in rocket science, which describes the relationship between a rocket's final velocity (= how long the engines are burning) and its mass fraction, or how much of the rocket is fuel vs. structure and payload. The longer you want to burn, the faster the propellant mass fraction increases until eventually you have no room for a useful payload, or even no room for your rocket.

In your case, although you're not using your engines continuously to produce a single final velocity, you're still burdened by how long you need to continue using them. (This is particularly true in the case of a hovering vehicle where you're constantly fighting against gravity. It would be less true, though still noticeable, in a ground vehicle that only uses fuel when moving.) The longer you want your vehicle to be able to travel without refueling, the greater percentage of it needs to be fuel, without limit.

So how do terrestrial vehicles like cars and airplanes escape this? It's simple: they make use of the atmosphere. Cars pull in oxygen from the air for combustion, making their fuel vastly more weight-efficient. Airplanes exploit the properties of the air to generate aerodynamic lift, reducing their thrust needs.

Given their other technological feats, your people might honestly be best off using combustion engines, then having processing plants recapture carbon dioxide and other combustion products from the air and process them back into fuel. This would require a lot of power, but it would offload the power needs from a small, inefficient mobile platform to a large, efficient stationary one. (In this case the gasoline or whatever is best considered as a type of battery.) Another option would be to carry batteries and have electrically-driven rotors or compressors - basically a very large recreational quadcopter.

As a final aside, having a lot of personal vehicles spewing xenon or other heavy gases everywhere might not be all that safe. Heavier-than-air gas will tend to pool in low places and displace oxygen, which can easily kill people.

As others have explained in detail, your limiting factor is how much fuel you can carry since there's a limit to how much you can compress a gas before you start running into problems. An indestructible tank won't help much, as the gas itself will condense into a liquid or start doing other strange things at high-enough pressures. The only way to make this remotely practical is if you can make your vehicle as fuel-efficient as possible.

When traveling by air, you'll spend more fuel fighting gravity than anything else. So don't fight it. An airship or aerostat of some sort could stay in the air without expending any fuel, so you can dedicate your onboard fuel to propulsion. Travel in the direction of the wind could be nearly free, albeit not that fast.

For a vehicle like this to be practical, you'd need to use the air that you're flying through as fuel instead of carrying a limited supply of it with you in tanks. You could have electric engines that suck in air and force it out a narrow nozzle on the rear of the vehicle. You're not moving at extremely high speeds, but at least it works.

I expect someone's going to point out this has massive scaling issues, but hey, you have near infinite energy, so let's swap things: instead of having a bunch of jets running over the surface trying to use compressed air, let's convert the surface into basically a giant air hockey table and make the transports just relatively light with solar sails. The transports have networked computers that connect to the transport system to indicate where they want to go, and then there are a series of "lasers" on the surface a scattered points to provide the horizontal thrust necessary (similar to the proposed solar sails for travel within solar system)

Dealing with any unintended consequences of the awesomeness of this setup are left as an exercise for the reader.

What you need to realize is that energy is $E = frac{v^2cdot m}{2}$ while impulse is $p = vcdot m$.

Each second, earth's gravity transfers an impulse of $p = gcdot m_ocdot 1s$ onto any object of mass $m_o$. If that object is to remain at rest (hovering in the air, or lying on the ground, doesn't really matter as long as it's not moving...), it must constantly get rid of this impulse. The object on the ground does so by transferring the impulse to the ground, an object in the air must transfer the impulse to the air.

Note that the last paragraph did not mention energy at all. The important figure is the impulse, only.

Now, look at the two equations for $E$ and $p$. The energy is actually $E = frac{v^2cdot m}{2} = frac{vp}{2} = frac{p^2}{2m}$. I.e. if you use an infinite mass, you don't need any energy at all (that's the object on the ground, which effectively uses the entire earth to get rid of the impulse). The smaller the amount of mass that you use, the more energy you need.

If you use your stored air only, you are in the worst possible regime: You are wasting Gigajoules of energy for nothing. If you just double the amount of air you accelerate by driving a simple, single stage turbine with your compressed gas, you have already doubled the lifetime of your fuel! The more outside air you accelerate, the longer your fuel lasts.

So, no matter how dense your pressured-air-energy-storage is, your vehicles will always suck in air from above and blow it downwards, simply because the fuel will last so much longer. You can basically assume that any hovering air vehicle will always use as much of its upper surface area for sucking in air as possible. Because, using only half the surface area means only half the lifetime of your fuel supply.

Btw, this is also the reason why helicopters need more fuel while hovering than when they are flying at moderate speed: The hovering helicopter can only interact with the air directly around and above it, which has already been accelerated by its hovering. The flying helicopter is constantly interacting with fresh air at rest, and thus distributing its impulse to more air. More air accelerated $Rightarrow$ air accelerated to lower speeds $Rightarrow$ less energy used.

Airplanes take this to the extreme: They maximize the amount of air they interact with per second by flying at 800km/h, and thus they minimize the resulting downwards speed of the air they leave behind. The burning fuel in the engine just turns a turbine which uses blades to accelerate as much air as possible backwards, which in turn moves the air foil to accelerate as much air downward as possible to be as efficient as possible. This double indirection is what makes current air planes as efficient as they are, enabling them to fly half-way around the earth without a stop.