Why aren't there cube-shaped suitcases or bags?

Because they are too unwieldy when you carry it on your side. And it’s easy to exceed the weight limits. I already can cram 23 kg in a single large luggage bag.

You can vacuum pack your clothes to make the most of the space in a luggage. Did you want to maximise your luggage space and thought cube shaped luggage might be a solution?

Because you only gain 11.7% volume and give up a bunch of other advantages, including making packing worse.

• The volume of luggage with maximum checked-luggage dimensions (W + L + H <= 62)^[1] is 27in x 21in x 14in = 7938in³.
• The volume of the cube is (20.7in)³ = 8869.7in³. Only 11.7% more.
• If maximizing (packing) volume (inside a large regular space) was the only design consideration, then yes it would be cubic (like balikbayan boxes as 200_success noted).

Now consider everything you lose:

• Much more unwieldy. Center of gravity and moments of inertia are worse (wonky, harder to pull with a handle on two wheels).

• Bulky: width of 20.7" is 48% wider. Blocks check-in lines, corridors, gangways.

• Areal density is worse, since the footprint 20.7² > 27 x 14. Worse in an elevator, or a departure lounge, or a car trunk.

• Although volume density and packing will be better in large spaces (e.g. warehouse, container), Volume density and packing are actually much worse, counterintuitively when either you pack it with other dissimilar-sized luggage, or in a small or non-cubic space

• You almost surely can’t stack two 20.7" cubes on top of each other in a car or taxi trunk. And hard to stack them diagonally.

• You can get a suitcase in a passenger seat at a pinch, but not a 20.7" cube (not in a medium-sized car, anyway)

• Packing density is worse e.g. in a car trunk. Having one identical dimension gives you fewer options for packing.

• As you say, it will be much less likely to fall over, but that’s actually a bad consideration for sliding on baggage carousels and loading ramps. Regular-sized luggage has lower static friction and slides over each other.

• Normalizing w.r.t. 20.7" sides, the normalized dimensions of regular luggage are (27/20.7, 21/20.7, 14/20.7) = (1.304, 1.014, 0.676). So you compromise on width to get more length, without losing much volume.

• Doesn’t preserve its orientation when loaded/packed/unloaded. Regular-sized luggage has a clear orientation, and can only be confused w.r.t. up/down (and even then we use labels or nametags or ‘Fragile’ tags to make that obvious).

  ^{[1] eBags has a very useful table of checked and carry-on luggage limits imposed by various airlines (US and international)}

The other, rather important point is this:
While that would indeed give you the maximum volume allowed, packing pretty much anything that fills the maximum volume is likely to take you well, well over weight. It’s easy enough to take a bag of a much smaller volume and go over the 18/20/30kg included weight limits. Extra weight gets expensive very fast, and if the bag is too heavy then the airlines may refuse to take it at all.

Balikbayan boxes, popular among Filipinos for bringing goods home when flying, are indeed quite close to cubic. Standard sizes are:

• Medium: 18 × 16 × 18 inches
• Large: 18 × 18 × 24 inches
• Extra large: 24 × 18 × 24 inches

While they are optimized for shipping, they are unwieldy for travel. In addition to issues mentioned in other answers…

• A cube does not offer choices of orientation for optimal packing into the trunk of a car.
• The wide footprint would prevent you from wheeling it onto an escalator (if it had wheels).

There is nothing stopping you buying the most cube shaped luggage you can find or have made and take it with you. I have seen ‘classic’ suitcases that are almost square, that used to be transported on ships. They are known, among other names, as cabin trunks.

The full size ones will be too big for nowadays flights, but you can get half size replicas and smaller ones.

They went out of fashion when people had to carry their own luggage, but you can revive the trend.

@Robert points it out, they would be too unwieldy. Although you, as a passenger, would have not to worry about them while being transported by plane, you still would have to get them from your home to the airport. And that’s assuming that’s the only moving around you’ll do. Will they have wheels and a long leash so that you can drag them around behind you? If you’d pick it up, you’d have to half outstretch your arm, which won’t be easy for many.

That said, somewhat close are things like this:

http://www.amazon.com/Plastic-Checkered-Storage-Laundry-Shopping/dp/B00MI9T7AA

A cube would maximise the volume while complying with the 62-inch limit

Maybe true from a mathematical point of view, but in reality cube suitcases would be:

• Harder to carry with a handle, you will have to stretch your arm away from your body, which will make things feel heavier.
• It will be harder to walk while carrying them.
• What about long objects you can easily put inside the normal suitcases?

It will also be much less likely to fall over

True, if it was the only suitcase. False in reality because it will be harder to place cube suitcases in containers on top of each other, this will make them much more prone to fall causing damage to them or to luggage handlers.

A disadvantage will be a great depth, with things packed many layers deep, so you’ll have to unpack before you can access things in the lower layers

This is true for one person, a person will have one or two suitcases, but a customs agent will have a hell of a job because scanners will not be that effective due to the many layers, and manual inspection will be difficult. That’s the reason why you have to put your luggage flat when you scan them, to make the scanner more effective. Making luggage condensed by default will be a big problem from security point of view.

When placed on a table or a bed in a cramped hotel room, it will take up less space on the table or bed.

I disagree, whatever space you save on one axis, you will get it in the other axis. Having a normal suitcase along the wall is indeed more space friendly than having a cube shaped suitcase with less space on the wall and much more space away from the wall.