The math is simple, its just the Archimedes principle, fluid drag force basics, and power law of propelers and jet engines.

Archimedes says:

m = V * (density air - density helium)

that is, the total mass that you can carry (including helium, propellers, fuel, all of it), scales linearly with the volume V of the balloon, and the difference in densities between the gas inside and outside.

There is little that you can do about the density difference beyond heating helium to make it larger, but at equal temp on the ground its 0.1, so you end up with:

m = 0.1 V

so if you want to carry 120'000 kg, then you need 1'200'000 m3 of helium.

That's a lot of helium. If you put it in a ball, the radius would be 66 m. or 132 m diameter. The wind surface of the ball would be 13'685 m2.

The force of air on a ball is obtained from the definition of the drag coefficient:

F = density_air * C_d * v^2 * A

The C_d is worse at low Reynolds number, so lets fix that at 1.0 (worst case, it gets slightly better as the flow speed increases, but whether its 1 or 0.8 it doesn't really matter).

So if you want to stay stationary at 20 m /s winds (72 km / h), then you need to apply a force of

F = 1.225 kg/m3 * 1.0 * 20^2 * 13'685 = 6.7 MN

The power required to produce similar thrust via a propeller is more or less

P = sqrt(F^3/(rho * A_propeller))

so with 10x 2m diameter propellers, you'd need

P = sqrt(6.7^3 MN / (1.225 * 10 * pi * 1^2)) = 2.8 MW

of power. If you actually wanted to move against the wind at 70km/h you'd need almost 6 MW.

That's not "a lot" but is not "nothing either".

If you take Rolls-Royce AE 2100 engine powering the C130 Hercules, each engine produces 7.5 MW. These engines aren't "silent", so riding in this thing isn't going to be nice, but no engine in this league is silent.

Considering that you also need electricity and other stuff on board for a crew of 20-40 people...

You end up that for 120 tons, you need at least one cargo aircraft propeller engine to be able to move at 70km/h against the wind (which is pretty slow). You might want two, you know, in case one fails, and the wind carries you to antarctic.

From those 120 tons, you need to subtract the helium, the propellers, the crew, etc. to get to the effective weight that you can carry.

If you wanted to go faster, notice that the drag increases with velocity square. You can try to reduce the surface like zeppelins do, but the surface is still going to be pretty big, because "volume" is what make these flight.

Also if you reduce the surface in travel direction, you are going to increase lateral surface cause that's how volume works, and that's going to make it worse when you have cross winds.

For any reasonable speed you might be looking at close to 20 MW of power, which is 777 jet-turbine like amounts of power.

So why doesn't make sense to build this to go from A to B?

Cause physics. If you want this to "float" (you can't really call this flying), you need absurd amounts of volume, and it turns out, that moving things with a large volume through a fluid requires a lot of energy.

You could do a more detailed analysis, improve the performance of every single component by 10x, but you can't change the physics.

Even if you just want to lift cargo, and barely move it from A to B, this wouldn't make sense, because the surfaces created by the huge volume would mean that you need absurd amounts of power to just stay stationary in case of "mild" winds. Even if you had enough power, wind changes direction quickly, but the huge propellers required for this don't. So if you need to lift heavy cargo, you probably need to position it with < 1mm tolerance, cause by definition you can't move heavy cargo after you position it, so you gotta get it right (e.g. something like a highway bridge).

You can't do that with this thing either cause of physics (big area, movement very sensitive to wind, impossible to control).

For the luxury thing, you end up much better with a luxury private jet, or a private 787 or whatever. The C130 Hercules, can take off with 137 tons, is orders of magnitude faster and cheaper, and only needs a crew of 3!

For transporting a bridge, you end up much better with a bunch of heavy cargo helicopters.

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Instead, if you just "float" somewhere, and you don't care were, balloons and zeppelins are pretty good.

They aren't as good as a modern glider obviously, but they benefit from working with the physics and not against them.

Other vehicles that "float", like boats and submarines, exploit the fact that water is way denser than air, yet even they have pretty big submerged volumes, but luckily for them water doesn't change direction as fast as wind does.