Because for any non-zero integers $x$ and $y$, $x^2 + y^2 \ge 2$, and the only possible values for $x^2, y^2 \le 3$ are $0$ and $1$, which cannot sum to $3$.

Because $3$ is a prime number of the form $4k+3$, and a prime of this form cannot be written as the sum of two squares.

Because modulo $4$, a sum of two squares can only be congruent to $0, 1,$ or $2$, but $3 \equiv 3 \pmod 4$.

Because the equation has no real solutions.