Ttyjukl

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.