A Cloak of Arachnida allows the wearer to cast the web spell, with the additional property:

the web created by the spell fills twice its normal area.

The area of an ordinary web spell is a 20-foot cube. What would be the shape and size of the area covered by this special enhanced web spell? Would it be a 40-foot cube? A cube with exactly twice the volume of a 20-foot cube? Two adjacent 20-foot cubes? Or something else entirely?

Twice the area of a 20-foot cube is two 20-foot cubes

I'm afraid this is one of those "There is no raw answer". So, we rely on a handful of "hints" from the devs until we get an official answer. The most relevant hint here to me is "plain text interpretation".

The book doesn't give any guidance I could find about doubling areas, at least not in this way. Normally, I'd pull up some definitions, but entries for words like "size" or "twice" aren't super helpful here.

Size (for example)

noun

the relative extent of something; a thing's overall dimensions or
magnitude; how big something is

The only course of action left is to decide upon the simplest and easiest to implement option.
Question: How many is one 20 foot cube?
Answer: One
Question: How many is twice of one?
Answer: Two

Two twenty-foot cubes... or a 20ft long, 20ft high, 40ft wide rectangular prism. Whichever.

I also realize that the spell says: (emphasis mine)

The web created by the spell fills twice its normal area.

_{An argument can be made that they're telling us to double the ground area, which supports my answer.}

The possibilities (from most to least likely in my view):

Twice the volume, but keep the height at 20

This is the most literal and simple to understand reading. This gives you either a shape that is 20x20x40, or if it is still a square, about 30 feet to a side (technically 28.28). This is also the same as doubling the area.

Twice the side length

We go from a 20 foot cube to a 40 foot cube. This is the easiest to envision, in my opinion. That gives us 64000 cubic feet of volume. You and I both know that 40 foot cube is much more than twice the volume of a 20 foot cube. But D&D is not a great reality simulator and it's an even worse math lesson.

Twice the volume (from the center)

A 20 foot cube is 8000 cubic feet. Twice that is 16000 cubic feet. The cube-root of 16000 gives us about 25 feet to a side.

Twice the side length, but keep the height at 20

20x40x40 gives us 32000 cubic feet.

Probably a 30-foot cube

(but possibly 25-foot cube depending on how we explicate the RAW)

If it means "map area", then it's a cube that's double the map area it occupies, and that's about a 28.3-foot cube (square root of double the area of a twenty-foot cube), which I would think a DM would round up to "30-foot square" map area.

But if you interpret "area" here as "cubic volume", then the answer is 25.2 (cube root of double the volume of a twenty-foot cube), which I would think the DM would round to "25-foot cube".

As DM I would incline toward the former rather than the latter interpretation, though both are defensible due to the ambiguity in RAW.

The Web's text states "fill a 20-foot cube" meaning, if you have a 20 ft cube (20 ft x 20 ft x 20 ft as per Cube PHB pg. 204) it will fill the whole area (area defined as a space, not a dimension).

If you cast Web that doesn't "fill a 20-foot cube", i.e. not anchored, or layered across a floor, wall or ceiling, then it will collapse into a dense 5-foot depth (20 ft x 20 ft x 5 ft). Subsequently, at the start of the next turn the spell ends with this result.

Cloak of Arachnida states "...twice its normal area..." (area defined as a space, not a dimension).

If the initial result fills a 20 ft cube "anchored" then twice the normal area (area defined as a space, not a dimension) the modified result would be 2 x 20 ft cube.

Since the area is space, then twice a 20 ft cube (in volume) is a 25 ft cube.

Unlike 3/3.5 walls, there is no option for changing the dimensions with Web. You only have a 20 ft cube to work with.