We do know the precise values of √2 and 1/3. They are, well, √2 and 1/3. We can also calculate their decimal representations as accurately as we need to as well. The fact that their decimal representations are infinite is more a consequence of how we write numbers down than anything else. For instance, if we used a based 3 number system instead of base 10, 1/3 would be written as exactly 0.1.