These are jujube fruits; they taste like apples and they are so full of nutrients and vitamins this title is well-deserved, (albeit emphatic). Available in fruit stalls here in Lebanon, they can also be found in the US and elsewhere in organic markets, Middle-Eastern stores, Chinese and other Asian markets, fresh or dried, and ordered online.
Jujubes were used in treatment of many illnesses in Chinese medicine. They are full of amino acids and antioxidants and help with many many conditions; they boost immunity, are a blood cleanser, soothing for the nerves, insomnia, anxiety; good for chronic constipation, diarrhea; lowers cholesterol; helps with fungus, bacterial infections, soothes the liver.
If interested, read:
as well as this article
and this Wikipedia article