Considering the human evolution and the diet of our ancestors, are humans biologically able to become a vegeterian and remain healthy?
It seems to me that "human evolution and the diet of our ancestors" is completely irrelevant. It's just a rephrasing of "if God had intended man to fly he would have given us wings".
Evolution is all about change and adapting to the environment. From an evolutionary point of view "because the ancestors did it" is pretty nonsensical.
Life evolved in the sea. Why are we on the land?! Moving from the oceans to land makes even less "biological sense" than a vegetarian or vegan diet. There are many other aspects of every day life where modern humans behave different from our ancestors (e.g. sexuality, violence, etc.).
A vegetarian or vegan diet is certainly possible, although there are some nutritional concerns (e.g. lack of vitamin D, B12) which can be easily addressed in a variety of ways.
A number of vegan athletes would find this question strange. Aren't they showing with their very accomplishments that it is possible to be vegan and healthy? Even to be vegan and deliver the best possible performance.
Also look at some vegetarian athletes that have been doing their best.
Paavo Nurmi, Venus Williams - big names.
When it comes to evolution and the diet of our forefathers, what are we talking about here? Are we looking at our teeth, that clearly don't belong to predatory meat eaters? Are we looking at our puny claws (also called 'nails') that won't let us rip apart the cutesiest rabbit?
Dr. Milton Mills makes a comparison of carnivore and vegetarian biology.
But there is no clear cause and effect in evolution. We evolved while eating meat and vegetables, we evolved while learning how to cook. Looking back won't necessarily give us the answers we need.
Your question is clearly very broad, but it can receive an answer.
Yes, one can be both vegetarian and healthy. Actually, healthier than average omnivorous person (if being healthy supports a comparative degree)
Other questions and answers cover in more details the short answer:
babies and young children (the most sensitive category when it comes to deficits) can be healthy as indicated in this question and answer
"designed" to be a vegetarian - this is covered in this question and its answers from here. I will include the conclusion and its references from my answer provided there.
Regardless of which option one chooses (being a vegetarian or omnivorous), nowadays we have more choices than humans could have in the prehistoric times: