This answers how to get modulo for a fraction but mostly rational numbers are addressed.
Therefore: $\frac{7}{8} \equiv 0 \pmod{7}$
What is the value of $r$ in the following Type of fractions? Is it defined?
$\frac{7}{\sqrt{8}} \equiv r \pmod{7}$
$\frac{7}{\sqrt{7}} \equiv r \pmod{7}$
I think your question isn't really about fractions with irrational denominators, it's about modular arithmetic for irrationals. The answer to that question is that it's not customary to write expressions like $$ \sqrt{99} \pmod 4 . $$ Modular arithmetic is usually reserved for integers, and for fractions with some integral denominators, as in the linked question.
In some computer languages and in some applications you might see $$ \sqrt{99} = 9.94987437107\ldots = 1.94987437107\pmod 4 . $$ The answer is the number between $0$ and $4$ that differs from $\sqrt{99}$ by a multiple of $4$.
Edit. As @lulu notes, you may want to think about the square root as an operation on the residues. Then for example $$ \sqrt{2} \pmod{7} $$ is $3$ or $4$ since each of those when squared gives an answer congruent to $2 \pmod{7}$.