Answer: No, it is not possible.
The size of an A4 paper is roughly 297mm by 210mm. If you cut 297mm in half (297/2), then 297/2=148.5 becomes your short side of the new paper. The 210mm side becomes the long side of the new paper.
If you check the ratio of the first sheet (divide 297 by 210) and the second sheet (210/148.5), you will find out that there is a small difference in the ratio between the two papers (the ratio is very close to the square root of 2). This means that achieving the wanted result with scissors and a ruler requires tremendous precision (more on this later).
A bit of research about international paper sizes leads us to consider the square root of 2: https://www.mathsisfun.com/geometry/paper-sizes.html
To obtain a perfect ratio between the two papers, we must follow a rule of international paper sizes. The rule in question goes as follows: if the short side of your paper is of size x, then the long side must be x*sqrt(2). If we take our initial paper 297mm by 210mm, we find out that 210*sqrt(2) roughly equals 296.98484 rather than 297. We should also note that the ratio ((x*sqrt(2))/x) is always sqrt(2). So, we can see that it would be very difficult to make a perfect A4 paper format by hand due to the precision limitations linked to using a ruler and scissors.
Now, repeating myself a little bit, if we divide our perfect 210*sqrt(2)mm by 210mm paper by 2 (divide the long side by 2), we will find that the proportions between the initial paper and the new paper are identical. The short side of the new paper would be ((210*sqrt(2))/2) and its long side would be of course 210mm. Dividing 210 by ((210*sqrt(2))/2) of course gives us sqrt(2).
Now, our proof:
- If A4(long side)=x*sqrt(2) and A4(short side)=x, for x an element of positive rational numbers, then A4(long side) is an element of positive irrational numbers given that a rational number multiplied by an irrational number=irrational number.
- Let a set "Ruler"={positive rational numbers from 0 to 30}. Given that A4(long side) is not an element of Ruler, then there is no way to accurately measure A4(long side) with Ruler. Thus, the only "solution" would be to round up or down A4(long side) to render it a positive rational number. But then, the ratio between the two papers would no longer be equal to each other (sqrt(2)), which can be demonstrated as such: A4(long side)/A4(short side)=A4(short side)/(A4(long side)/2)=(2*A4(short side))/A4(long side)-->(A4(long side))^2/(A4(short side))^2=2-->A4(long side)/A4(short side)=sqrt(2). If we assumed that A4(long side) and A4(short side) are positive rational numbers, then there is a contradiction that A4(long side)/A4(short side)=sqrt(2), given that the quotient of two rational numbers cannot be irrational. Therefore, A4(long side) must remain irrational to retain the ratio.
- We conclude that A4 paper formats can't be made with a ruler and scissors.