I don't know the minimum, but I can tell you that I have bred them in about 350 gallons. This was in a big round 500 gallon tub, about 2/3 full. I have not bred jumbo koi in this tub, but for koi up to 25" (64 cm) it has worked well. It probably depends on the size of the koi.
When I bred my showa, i built a 3ft x 4 ft x 2ft high for a spawning pond. I have 300 gallon rubbermaids tank, but I didn't use it for breeding purpose. I had raccoon problem. It comes once per week, so I don't feel safe to put 24-25 inch koi in a 300 gallon rubbermaids tank.
You can breed koi in 350 gallons of water like Matt stated above and I've also used the same method but for a novice the problem comes when you have to start addressing water quality issues with that small amount of water. So if I was you I would find the largest reservoir that you can in order to make your life easier and give those fry the best chance.