Decompose 96 into two factors whose sum of squares is 208.
As the product of the two numbers is 96, they are divisors of 96. Factoring 96 into prime factors, we find $96=2^{5} \times 3$, so its divisors are:

$$
1,2,3,4,6,8,12,16,24,32,48,96
$$

The divisors 96, 48, 32, 24, and 16 do not work because their squares are already greater than 208. This leaves us with

$$
1,2,3,4,6,8,12
$$

whose squares are:

$$
1,4,9,16,36,64,144
$$

Now we see that the only possibility is $64+144=208$. Since $8 \times 12=96$, the numbers are 8 and 12.