My understanding of your rules:
A. Transport two of every (non-fish) vertebrate to Iraq.
B. Put them on a boat that would fit them all.
C. Do this using only historically reasonable technology, with no additional divine intervention.
Rule C is important, because once we allow magic, what's the point in asking the question? Any omnipotent deity who really cared could have poofed the boat and animals into existence instantly and skipped the entire silly story. I will, however, assume that the animals are willing participants in this process though - that once Noah reaches their range, he won't have to find and trap them. If not, I'm fairly certain that the process couldn't be completed by Noah and his family ever.
Building the boat would have required a substantial amount of time. Some creationist types have put construction time at between 70 and 120 years. Recall that Noah and co. were supposed to have lived for several hundred years. There are massive logistical problems with building the biggest wooden vessel ever constructed in ancient Iraq, not the least of which is supplies. The sheer amount of wood required to build the vessel is enormous - it would have had to have been imported to the area from elsewhere.
Most of the vertebrate species on Earth are quite small. For instance, the vast majority of the 6000 amphibian species would, if placed in containers roughly the size and shape of 10 gallon aquariums, fill a volume of about 120'x15'x8' (40 3' wide shelves that are 15' wide and 8' high, crammed full of aquariums with just enough space between shelves to walk. If you could just throw all the frogs in a bucket together, then you can collapse this down substantially, of course. All but a few of the amphibians would fit into such a container easily.
The other groups of vertebrates are similar, with most members fitting comfortably in a breadbox sized volume, but there are as many as a thousand species that need considerably more room. It's also difficult to count how many species there really are. For instance, the number of extant Equus species (horses and friends) is anywhere from 7 to 24, depending on how you count them. Monitor lizards dragons, the big cats, anacondas, pachyderms, and large primates are all problematic in this regard, and there's no good way to count them up by volume. If we give each of them a goat-sized stall (5'x8') and assume that the ones that are a little smaller will average out with the ones that are a little bigger, then considering all of the larger animals on the planet, you're looking at something like 40,000 square feet of floor space. With enough room to walk between stalls for feeding, the big animals would fill up about one and a third entire floors of the biblical ark. The non-fish vertebrates would have fit into the ark, barely.
I initially thought that packing sufficient food for the trip would be problematic, but it is less so than I originally thought. The elephants would eat about 500 pounds of high quality food per day. The voyage was roughly a year long (a bit more if you add up all of the days) so that is roughly 70 tons of elephant food alone, another 70 cubic meters of space (assuming the food is packed extremely efficiently.) Food for a similar duration of the trip for all of the other animals would be problematic to collect, and for herbivores, the food will take up a substantial amount of space as well, though there is no reason you couldn't put that food in the enclosure with the animals. Gaining access to sufficient quantities of fresh water for all of the animals would be a challenge though.
Now, could you cram all of the animals into smaller enclosures and have plenty of room? Of course you could, and if we're assuming divine intervention to deal with all of their behavior for the collection and trip, it might be appropriate to do so. In any case, the biblical dimensions, perhaps with some additional height added so that we can get three floors into it if needed, should be sufficient to hold the vertebrates and a year's worth of food.
Using your rules, it would take a very, very long time and be exceedingly difficult. Any voyage would have been done by sea or caravan overland, and the logistics of moving that many animals to Iraq are daunting at best. Basically you'd have to build arks just to collect the animals to go on the ark. That is going to take years itself.
The bigger problem is speciation. New species arise at a rate of roughly 1-3 per million per year. We'd expect as many as one new vertebrate species each decade. Columbus's trip took about a year, with essentially no overland travel involved. Magellan's crew got back after 3 years, again without significant overland travel. Lewis and Clark took 2 years to walk a single line across the continental US. Each new species that arose would set Noah back by years, and the time required simply to sail to every single required landmass on Earth would be astounding. I honestly don't know how to put a number to it. You also introduce the horrible logistics of caring for enormous numbers of animals while you wait for Ham to get back with the latest variety of leopard frog or whatnot.
This amount of time is entirely dependent on the level of magic we're willing to accept. If the animals are not cooperative, then the collection will take forever due to speciation. If the animals are super cooperative, traveling to easy to reach ports for collection, then it might just be possible over the course of several decades or centuries.
