Is it possible to construct a square with an area of 7 units?
We know that $x^2 = 7$, and therefore, $x = \sqrt 7$.
But, as $\sqrt 7$ is an irrational number, how can we be sure of the length $\sqrt 7$?
Also, how do we prove $\sqrt 7 \cdot \sqrt 7 = 7$?
What do you mean by construct? If you mean with a straightedge and compass, the answer is yes (see this question).
We have that $\sqrt{7}\times\sqrt{7} = 7$ by definition of $\sqrt{x}$.
1) straight edge and compass constructions allow us to construct circles and lines and their points of intersect. As lines are of the form $y=mx+b $ we can construct all rational numbers. As circles are of the form $x^2 + y^2=r^2$ We can construct irrationals that are some combination of square roots.
So the square root of 7 is not a problem. (We construct a right triangle with sides 2 and 1 and the hypotenuse is root (5). We construct a right triangle with sides 1 and root (5) and the hypotenuse is root (6). We construct a right triangle with sides root (6) and 1 and the hypotenuse is root (7).)
2) $\sqrt {7}\times\sqrt{7} =7$ by definition. We define $\sqrt{7} $ to mean the positive number when multiplied by itself equals $7$. The real question is how do we know there is such a number and that it is unique.
We know such a number exists because $2.5^2=6.25 < 7$ and $2.7^2=7.29 > 7$. We know that numbers are a continuum and between any two numbers we call find an infinite of others of every intermediate value. Between $2.5$ and $2.7$ there are real numbers for all quantities and their squares can describe all intermediate values between $6.25$ and $7.29$.
It's a naive misconception, to believe that to "know" a number we must be able to express its decimal representation. A decimal representation is useful for writing things down but it has absolutely nothing whatsoever to do with us "knowing" what the number is.
By the pythagorean theorem we know that if we construct a square with sides 1, than the diagonal will be exactly $\sqrt {2} $. That we will never be able to measure it with a rational ruler is utterly irrelevant. We still know exactly what the number is even though we can never write it as a decimal. That is because writing it as a decimal has nothing to to with knowing what a number is.
If we take a square whose sides are the diagonal of a rectangle with sides one and the diagonal of a rectangle with sides $2$ and $1$ then that square will have area $7$ and it's sides will be $\sqrt {7} $.
