While Dino DNA could only have been extracted from blood inside mosquitoes that were preserved in amber to suit the story, it is in fact possible for us to get material from recently extinct animals to try and bring a species back. That is where science fiction ends and science fact begins.
First off, the material must be extracted from an animal that has been dead for a maximum of 5 days, or if longer, the animal must have been preserved by being frozen immediately after death to preserve its cellular integrity.
In order to bring back a rare Asian ox, the Gaur, Scientists infused a regular cow's egg with the genetic material of a living Gaur. While the egg was accepted by Bessie the cow's immune system, the Gaur calf suffered the same fate as all of the other cloned animals, and it succumbed to illness shortly after being born.
If the technique had been successful, it could have been used to repopulate extinct species, or strengthen the ranks of those on their way to extinction.
If implanting eggs which were taken from a host animal with the DNA of another animal for the purpose of its acceptance by the host body's immune system produces similar results every time, maybe the implanted embryo's immune system fails to develop as that of a normal fetus. Perhaps by looking more closely at the immune system and its adaptability, we will understand why genetically identical clones are always inferior to the originals.