I can't think of an instance that would make this true since it seems to me that if a,b; b,c; a,c are independent of one another, so would be the case of a, b, c
$$P(A\cap B) = P(A)P(B)$$ $$P(B\cap C) = P(B)P(C)$$ $$P(A\cap C) = P(A)P(C)$$ $$P(A\cap B\cap C) \neq P(A)P(B)P(C)$$
Let $X_1, X_2$ be two independent rolls of a 6 sided dice. Let $A$ be the even that $X_1$ is even, $B$ be the event that $X_2$ is even, and $C$ be the event that $X_1 + X_2$ is even.
You can check that each pair of $A, B, C$ are independent but the three of them together are not independent. Intuitively, knowing $A$ and $B$ tells you a lot of information about $C$.
Space has $4$ points $(i,j,k,l)$ each with prob. $=\frac{1}{4}$.Let $A=(i,j)$, $B=(j,k)$, and $C=(j,l)$. Then $P(A)=P(B)=P(C)=\frac{1}{2}$. $P(A\cap B)=P(j)=\frac{1}{4}$, $P(A\cap C)=P(j)=\frac{1}{4}$, and $P(C\cap B)=P(j)=\frac{1}{4}$. However $P(A\cap B\cap C)=P(j)=\frac{1}{4}$
