Right rectangular prism has 12 edges: 4 edges in x-direction;
4 edges in y-direction and 4 edges in z-direction.


Let  "a"  be the length of any of four edges in x-direction;

     "b"  be the length of any of four edges in y-direction;

     "c"  be the length of any of four edges in z-direction.


We are given that 

    2ab + 2bc + 2ac = 1     (1)  (The total surface area of the prism is 1 square unit)


We also are given that

   4(a + b + c) = 2              (the sum of all the edges of the prism is 2 units)

so

   a + b + c = 1/2.         (2)


Square equality (2)  (both sides) .  You will get

    a^2 + b^2 + c^2 + 2ab + 2bc + 2ac = 1/4.


Replace here 2ab + 2bc + 2ac by 1,  based on (1).  You will get then

    a^2 + b^2 + c^2 + 1 = ,

which implies

   a^2 + b^2 + c^2 =  - 1,

or

    a^2 + b^2 + c^2 = -.


Since the left side must be positive as the sum of the squares of real numbers,
while the right side is NEGATIVE, such equality is not possible - it is SELF-CONTRADICTORY.


From it, one should conclude that the described situation, as it is given in the post, is IMPOSSIBLE,
and with given data, this problem HAS NO SOLUTION.