Step 1: List what a single digit can be.
A digit of a number is one of $0,1,2,3,4,5,6,7,8,9$. Since the product of the three digits is $70$ (not $0$), none of the three digits can be $0$; all three must come from $1$ to $9$.
Step 2: Try the largest possible digit first.
$70$ is fairly large for a product of three single digits, so check if $9, 8, 6$ divide $70$ evenly. $70 \div 9$, $70 \div 8$, and $70 \div 6$ are not whole numbers, so none of $9, 8, 6$ can be one of the digits.
$70 \div 7 = 10$, so $7$ can be a digit, leaving the other two digits to multiply to give $10$.
Step 3: Split the remaining product.
We need two single digits whose product is $10$: $2 \times 5 = 10$ works, and both $2$ and $5$ are valid single digits.
Check there is no second way: $1 \times 10$ fails since $10$ is not a single digit, so $2$ and $5$ is the only valid pair.
Step 4: Confirm the full set of digits.
The three digits are $2, 5, 7$, and indeed $2 \times 5 \times 7 = 70$, matching the question exactly.
Step 5: Add the digits.
$2 + 5 + 7 = 14$
This is the sum of the digits of the three-digit number, matching option (B).
\[ \boxed{14} \]