s = 29, {p; q; r} = {2; 3; 5}

Let p, q, r, and s be solutions to the problem and WLOG as the equation is symmetric for p, q, r let $p \leq q \leq r$. As s > 2 is prime, then it is odd which means that there is an even number among p, q, and r. Therefore p = 2. One of the numbers q, r is a multiple of 3 (otherwise $p^4+q^4+r^4+119$ would give remainder 2 modulo 3, which isn't a quadratic residue modulo 3). Hence q=3. Knowing p and q we see that r is divisible by 5 (otherwise $p^4+q^4+r^4+119$ would give remainder 2 modulo 5, which isn't a quadratic residue modulo 5). Hence r=5. All that it's left to do is to check whether s is a prime number: $2^4+3^4+5^4+119=841=29^2$