You are on the right way:

Suppose, for contradiction, that there is a rational number a/b with gcd(a,b)=1 whose square is 98.

(a/b)^2=98

a^2=98b^2

(a^2 / (49 b^2) = 2

(a/7b) = sqrt(2) (or (a/7b) = - sqrt(2))

(a/c) = sqrt(2)

=> sqrt(2) is a rational number

=> Contradiction.