No, it's not possible (rather it's possible but not desirable), nor is it natural. The natural thing for ruminants (and humans for that matter) to do, is wander freely from place to place with no limits or boundaries, allowing vegetation to get eaten down (or be left alone) by succeeding populations of all sorts of animals, insects, microbes, etc. Then maybe them coming back through the same spot 8-36 months later, etc.etc.etc. Unfortunately, humans have chosen to abandon this nomadic lifestyle. I think ultimately this is the root cause of our climate crisis, but obviously not one that we will ever be inclined to address.
Many of my dairy animals live through their long natural lifespans; it hurts to see my 14 year old best friend (usually the matriarch of the herd) get torn apart by a pack of coyotes though, or suffer other disease or mechanical injury as a result of old age and bad luck. However, on a large plot of land (say 60-100 acres+) that is not managed, they will surely run out of nutritionally dense foods to eat, as they will freely pick and choose certain stands of certain forbs or grasses, and let others lie dormant (these ultimately going to forest). The result is the highly nutritious choice foods get eaten to death, forest slowly sprouts everywhere else, and animal health suffers as a result. Now, if you had a large mixed herd, it becomes a possibility (think cows, goats, sheep, pigs, chickens, other fowl all living together). Now, if you can take that same large plot of land, and that same mixed herd of various animals and preferably plants, insects, and fungus, and rotationaly graze in much smaller paddocks, you can make magic happen, assuming your soil was dense enough in minerals to keep the animals happy and healthy on pasture alone (unlikely in many parts of the world without mineralizing the soil first).