The previous answers contain a lot of data regarding idividual animals, including human beings, so I won't repeat it all here. However, I will posit that not near enough information is available on this subject to make a true assessment of endurance, especially when compared to humans.
For instance, there is no mention at all about the conditions under which such an endurance test is run. Do both the "competitors" have access to water? How hot is it when they perform at their highest capacity, and does that highest capacity actually occur in extreme terrain or extreme cold temperatures? Humans operate in various temperatures, but typically not without clothing or skin protection, which no animal needs. As homo sapiens are not all created equally physically, the nationality of the human runner could have a major role to play. Those born in Africa (an average adult male) are best equipped for surviving and enduring in the hot arid savannah sun, whereas an average shorter stature Japanese man who has much fairer skin might suffer mightily in the brutal heat. So would an average man from extreme northern climes where stature can be even shorter (to conserve heat) and the temps plunge below zero regularly.
Conversely, animals do actually make up for their lack of single-sprint endurance by using speed, rest, speed, rest rhythms. By way of an example, an antelope that can run at say, 45-50 miles per hours will have covered that distance in sixty minutes. Then he can stop, rest, get his breath, take a drink, etc. The human, on the other hand, will only have made a max of 20 miles in that time at a run.
The human will get no rest, no fluids and no respite if he is to pursue the antelope. He is at less than half the distance when the antelope can stop and rest. If the antelope rests and refreshes for a half hour, he can resume his run and go for another hour, putting another 45-50 miles behind him, while the human is slogging along without rest.
When the antelope again rests, the man has covered 40 miles in two hours and had no water at all. The antelope is now almost 80-100 miles along, and can rest for longer periods and even stop to graze as each period of running passes because the human falls further and further behind.
To my knowledge, there is no data to suggest that such numbers are impossible.
Animals are built from birth to run in bursts, whereas an athletic human must train very hard to attain the ability to run long distances. Thus, an average antelope is more than able to best an average human. It therefore stands to reason that it would take an extraordinary human to even begin to call it a contest.
While animals use the "run, pause, run" technique, it is what has kept them out of the jaws of predators and under extreme conditions. Humans have none of that conditioning on average or by birth.
Just as there are horses best suited for the grueling trials of the steeplechase or the Triple Crown, there are also "average" horses that are nowhere near able to perform the feats the trained and bred thoroughbreds can do. Wild animals must be at their peak at all times or they become food for something else. Humans just don't have that incentive and therefore it would take a human of extreme physical conditioning to give chase to a wild animal with any hope of success. This all means that the question of which has more endurance is skewed from the outset. It is not a valid test if one subject is outside the average of the species. We may not like it, but the average human is pretty handicapped in the wild. We have to tilt the circumstances to give ourselves a chance to win at all in such a contest.
I have no problem with the notion that humans could indeed run down some animals on foot and unaided. Even the average man could do it, up to a point. But I can also easily see a man being left hopelessly in the dust behind some of the efficient wild animal runners.
Barring data from an actual comprehensive test using controlled conditions and subjects, I will weigh with this: Humans are good and very adaptive, but we are no match for those who were born to the task of running to survive.
the previous answer stated that the kangaroo has the best endurance.... they can stay at full speed for twenty miles hopping 15 to 20 feet at a time.
However, recent studies suggest humans actually have the best endurance. We may not be the fastest, but we don't need to be. Our leg muscles have evolved to require a minimum amount of energy to pick up speed, whereas all other land animals need ALOT of energy to go from a trot to a gallop. Plus, we can expend heat even while running (through sweating and breathing through our mouths), whereas most other animals disperse most of their heat through panting, which they can't do while running. It allows us to run at our top speed for far longer than animals. Even to this day, there are tribes of humans who hunt simply by running their prey to the point that the animal collapses from sheer exhaustion.
We can even run a horse to death. By the way, the horse stat listed below? It can only run that long if forced to by their rider. Otherwise, they won't. No animal, other than humans, will run at top speed nonstop. They will stop to rest occasionally. We don't need to.
Another, often overlooked "marathoner" is the Bar-tailed Godwit (Limosa lapponica). It flies for 7 - 8 days with out stopping, eating or sleeping. The voyage from Alaska to New Zealand spans approx. 7000 miles (11,00 km).
Well if you want to look at land animals it would be
Pronghorn:50-55mph for 30 minutes
Arabian Horse:35-40mph for 4 hours
Ostrich: 45mph for 30 minutes
Husky: 25-30mph for 6 hours
Brown Hare: 40mph for 1 hour
Humans: 12-20 mph for 16 hours or more
In the air it is
Canadian goose: 50mph for 12 hours
Bar-tailed godwit: 50mph for 150 hours
Arctic tern: They make the longest trip of any bird traveling 11,000 miles one way, but it is undecided if they make stops or not. they have a average travel speed of 50mph
Albatross: 75mph for 8 hours