How can we construct rectangle with non-zero area when all its lines have zero area? [Content was updated after discussion with David C. Ullrich]
I have one method of construction of rectangle, but it gives incorrect results. The reasoning for this faulty method is following:

Consider this rectangle:

As you can see I divided it into three parts. Obviously, I assume
  (because it seems self-evident to me) that if we sum up area of each part
  then we will get area of the whole initial rectangle.
I also assume (because it seems self-evident to me) that if I made the
  blue part little more narrow it wouldn't change this fact. If I
  separated the rectangle into 100 equal (and consequently - very
  narrow) parts, it would be true too nonetheless. BUT ...
But what if we divide the rectangle down to even smaller parts? What
  if we will take all infinitely many LINES that make up the rectangle?
  Now things become strange. An area of each line is exactly equal to
  zero. And infinite sum of zeros is equal to zero.But this is false,
  the rectangle clearly has area greater than zero! In other words, we
  took all parts and the result turned out to be LESS than the whole!

This question was marked as duplicate of another question (How can points that have length zero result in a line segment with finite length?), yet I'm afraid that I don't quite get the answers for said question (and for If point is zero-dimensional, how can it form a finite one dimensional line? too)
My mathematical background: Russian school course of mathematics (including calculus) + predicate logic + linear algebra + graph theory + some basics of set theory (still learning it) + some basics about statistics and probability (still learning them). All school mathematics I studied with a private tutor, everything else I was basically self-taught. This being said, I feel rusty in all it (except predicate logic and set theory), but I think I will be able to quickly relearn concepts from said subjects if it's needed to understand explanation.
 A: It's  not necessary to mention measure theory to explain what the hole in your logic is. You're assuming this:


Assumption If you divide a rectangle into infinitely many disjoint pieces then the sum of the areas of the pieces is the area of the rectangle.


How do you know the Assumption is true? You can't know it's true unless you know how to prove it, and you have not proved it.  I know you haven't proved it because it's false - I  know it's false because the example you give, dividing a rectangle into lines, shows that it's false.
And that's really all there is to it - you're relying on an unproven assumption, and that assumption is simply false; if you assume something false you can prove any crazy thing you want.
What it comes down to is "it simply doesn't work that way". That's not the answer you want, but it really is the right answer. Why doesn't it work that way? Well that "why" is  problematic. In fact it does not work that way, and the example you give shows it doesn't work that way.
Comment You say "I don't have any definition. I understand this concept intuitively". The moral here is that your intuitive understanding is simply wrong. When you start talking about infinite sets a lot of intuition turns out to be wrong - your intuition is based on examples of finite phenomena.
Mathematical truth doesn't care about your intuition. It's nice when the two agree, but when they disagree that's just too bad for your intuition.
Meta-Comment You say "There must be something wrong with my logic". But there's no "logic" at all here - you're talking about your own personal intuition, which is going beyond logic.
Comment About measure theory: Measure theory is certainly relevant here - it explains conditions under which modified versions of the Assumption are true. But it's not needed to explain the error above - "it simply doesn't work that way" really is the right explanation.
Aside, not meant for the OP but for readers who believe that the answer to the question really does involve measure theory:  Presumably we're claiming that the right explanation is something like this:


Purported explanation Lebesgue measure is countably additive, but the partition of the rectangle into lines involves uncountably many sets.


Well, no.
(i) That's not an explanation of why the Assumption is false! It's an explanation of why the Assumption does not follow from measure theory;  of  why its falsity is consistent with measure theory, not at all the asme thing as an explanation of why it's false. An explanation of why it's false consists of a counterexample - in fact the OP gave a counterexample.
(ii) It's also not an explanation of the error in the OP's reasoning. It might  be an explanation of that if the OP's reasoning  involved measure theory.
A: Maybe you're right!
You've discovered a paradox. All of the following seem to be true, and yet they contradict each other:


*

*A rectangle has positive area.

*A rectangle is made entirely of line segments.

*A line segment has area 0.

*If a "whole" is made entirely of pieces, then the area of the whole is the sum of the areas of the pieces.


In order to resolve the paradox, we have to deny at least one of the four premises. The standard definition of the word "area" is one that rejects premise 4 and admits the other three premises.
However, we could define "area" (or even "rectangle") differently and find a resolution that rejects any one of these four premises (and admits the other three). All of these resolutions are self-consistent, so they can all be considered "correct", but they are also usually considered to be "worse" than the standard definition of the word "area".
Denying premise 1
If we want, we can deny that a rectangle has positive area at all; we can define the word "area" so that all rectangles have area 0. But this definition of the word "area" doesn't match our intuitions about area at all, so let's move on.
Denying premise 2
An interesting resolution is to deny that a rectangle is made entirely of line segments.
It's hard to make this one work, because one of the core assumptions of standard geometry is that all "shapes" are made up of points, which entails that all rectangles are made up of line segments. However, if we make some serious changes to our fundamental assumptions, we can end up with a theory where rectangles are not made entirely out of line segments, but rather out of rectangles with infinitesimal (but non-zero) width.
I don't actually know of any formulation of geometry that works this way, but from what little I've heard about smooth infinitesimal analysis, it might be such a formulation.
Denying premise 3
We can deny that a line segment has area 0. This resolution doesn't seem to require us to change any of the fundamental assumptions of geometry; all we have to do is change the definition of the word "area".
All we need to do is invent a new definition of the word "area" under which premises 1, 2, and 4 above are all true. There are only two problems with doing this:


*

*I don't know whether or not anyone has actually done it yet. Maybe there is no easy way to do this after all.

*The new definition of "area" can't be a real number, because there is no real number which can be added to itself infinitely many times to produce a result which is positive and finite.


Denying premise 4
This is the resolution that results from the standard definition of the word "area" used nowadays. Under this definition, the area of a line segment is exactly 0. The only "problem" with the standard definition is that the area of a "whole" can be greater than the sum of the areas of the pieces, if there are uncountably many pieces.
In other words, the standard definition of "area" is not additive. Intuitively, it seems like area should be additive. After all, you said that it seems like self-evident truth that if you divide a rectangle into pieces, then the sum of the areas of the pieces is the area of the rectangle.
Combining 3 and 4?
I have a resolution you might like.
It seems like if you take a rectangle that has area, and divide it into equal pieces, then those pieces ought to have a nonzero area. So go ahead and define the "true area" of a shape in whatever way you want so that this holds true. If you think that a line segments should have an infinitesimal area, let's go ahead and say that the "true area" of a line segment is an infinitesimal number.
Meanwhile, define the "formal area" as the standard definition of area used by mathematicians today.
If you do this, then the resolution is simply the fact that "formal area" is not the same thing as "true area"! Instead, "formal area" is merely the closest real number to the "true area".
So although the "formal area" of a line segment is 0, you can think of this as being a rounding error, and the "true area" is actually greater than 0. Once you bring uncountably many line segments together, then the rounding error becomes significant.
But keep in mind that when mathematicians say "area", they mean what I'm calling "formal area" here. My so-called "true area" isn't something that mathematicians nowadays often think about or talk about.
