Simple:
* The statement "This statement is false" is a classic example of a paradox.
* Her life was a paradox: she was both incredibly successful and deeply unhappy.
Figurative:
* The peace treaty was a paradox, bringing an end to the war but leaving the country in a state of perpetual unrest.
* Love is a paradox: it can be both the greatest joy and the deepest sorrow.
With emphasis on the contradictory nature:
* The paradox of the internet is that it connects us while simultaneously isolating us.
* The paradox of happiness is that the more you chase it, the less likely you are to find it.
Using the word "paradoxical":
* It was a paradoxical situation: he needed to save money but also needed to spend money to make money.
* The results of the study were paradoxical: the group that exercised less actually had better cardiovascular health.
Remember, a paradox is a statement or situation that seems contradictory but may be true. Choose the example that best fits the context of your writing!