Take a tooth. Leave it in a cave for 5000 years. Retrieve it and examine the tooth: after all that time, those seasons passing and bacteria working away, what is left of the original animal? Not a lot; but not nothing.
There remains still some DNA from the original owner of the tooth, but degraded, fragmented into little pieces, and overwhelmingly outnumbered by the DNA of all the bacteria that have grown and reproduced and died in the tooth. Finding the DNA of the original animal would be like finding a needle in a haystack—if the haystack was really big and the needle was also a piece of hay, just slightly different from all the other hay.
And yet: we can do it.