jmc

The base 4 number $b_k{b}_{k-1}\dots b_2{b}_1{b}_0$ is equal to $4^k{b}_k+4^{k-1}{b}_{k-1}+\cdots +16b_2+4b_1+b_0$, so when this number is divided by 8, it leaves the same remainder as when the number $4b_1+b_0$ is divided by 8 (since all higher terms are divisible by 8). Hence, when the number $120301232_4$ leaves the same remainder as $32_4$, which is equal to $4\cdot 3+2=14$. When 14 is divided by 8, the remainder is $\boxed{6}$.