Is the area of a line really $0$? Let's take a square with side $A$. The area of this is defined as $A\times A$.
The way I explained this to myself is by reapeatedly deviding the square till I reached a single line.
Then stacking up lines one on top of the other $A$ times.
But if the area of a line was really $0$ no matter how many lines were stacked up, they would always have an area of $0$.(would exist in one dimension only)
Another way to see this is defining a function $V(F)$ which gives the volume of two dimensional figure $F$.
If this volume became $0$ , no matter how many such Figures I stack up on each other they would never have a volume.
When this is combined with physics it makes a little more sense.
Even if I were to chop up a cube infinitely I could never reduce its height to less than the Planck length. This means that the volume would be $Area \times  ℓP$ (Planck length)
So the question is for a three dimensional object to exist must every two dimensional object have a non zero volume?

Your interpretation of the area of the square makes sense if each of your "lines" has a width of $1$ (that is why you have exactly $A$ such lines). Usual lines have zero width and you need to have uncountably infinitely many of them to obtain a height of $A$.
