Keep an eye out at Privoz Market — you're bound to uncover a great travel trinket in one of its many stalls. After you've had a good look at the wares, try some fare from one of the local food vendors.
Situated in Odesa City Center, this community marketplace is worth a look. But you'll find loads more to see in the area. Arcadia Beach, Odesa National Academic Theater of Opera and Ballet, and Deribasovskaya Street aren't too far from Privoz Market.