Eustress is actually important for us to have in our lives. Without it, we would become depressed and perhaps feel a lack of meaning in life. Not striving for goals, not overcoming challenges, not having a reason to wake up in the morning would be damaging to us, so eustress is considered 'good' stress. It keeps us healthy and happy.
Eustress also reminds us that we can view many of the stressors in our lives as 'challenges' rather than 'threats', and have extra vital energy to handle these stressors, without a feeling of being overwhelmed or unhappy. (Read this for more on threats vs. challenges, and find ways to change your perspective.)
That said, while eustress doesn't generally carry the same type of damage as chronic stress, too much eustress can still tax your system. That's why it's important to have 'down time' and balance in one's life. Read this article for more on finding balance, or this one on the importance of fun.