OMG fun question! And based on your comment, I can put them in shipping containers (does that sound cruel?).
Float the Bronx Zoo. We can speculate that the visitor attractions, access, and exhibitions can be eliminated, but utilities have to be inserted onto the Ark. It will need access to electricity, water, stormwater drainage (a must), and wastewater.
If Noah is smart and can create a ship that is multi-leveled, he can follow the example of the Maersk Class container ships. Afterall, it is only for 40 days that the animals must be contained. There are 650 species (x2 genders), meaning 1300 animals. Not too difficult.
Many, such as the world of birds exhibit can be kept together, while others are separated. But you don't need a full savannah for an elephant couple for just 40 days. The largest enclosure can be one shipping container, while the smallest will be terrariums or aquariums (do we really need to provide for the salt-water fish?).
So let's do 300 large containers, an aviary, and a room for 300 small-animals. This includes some separations since obviously not all genders go together at all times unless they're mating.
We can easily fit this selection of the largest containers onto a container ship, as well as your reptile, bird, bug, etc. rooms. In fact, the Flextowe can hold 19,000 making heaps of space for you to add your accessories.
In an Urban Planning masterplan (I know it's a different science, but..) we allow for 15% for utility production, distribution, and access. You will want to shy away from solar, but wind and generators should be fine. A power/desal-water station takes about 1ha of space, which, even with weight is nothing for your mighty Flextowe. Fuel for 40 days is dependent on your location, but a good rule of thumb for a home (x20 homes?) is 10kWh. Tiny little primary substation. We don't really need huge spaces for cars, so, while I can't do the math-thingy's:
300++ containers (11m x 2m x 2m) = 16,000m3
+15% for access, ventilation, utilities = 18,400m3
This easily fits on the Felixtowe with lots of space to spare. This is where you can set your wind turbines, generators, water storage (I suggest rainwater collection...) and even Noah's house, as you like.
Concern 1: Employment - 40 rainy days of feeding, mucking, healthcare, etc. requires way more than a family. I can't find statistics that separate researchers from day-to-day employees, nor from public-interface employees, but I suspect it's still a lot of people. Next you need help loading, unloading, and distributing the animals.
Concern 2: Sex - Two animals of a different gender might not engage in sex ever. Not just gay animals, but simply animals that are picky.
Concern 3: Return - Noah left at one point, and then presumably landed on Mt Sainai. So... how should he and his family separate the lions from the gazelles way up there. And who is handling the Butterfly Garden or the Bug exhibit?