Barna is the definite plural form of the word barn, and it means "the children". Because it is plural, you'd have to use mine rather than mitt, producing the sentence Barna er mine.

Edit: Also, "barna" ≠ "barnet". This may be confusing because "-a" is used as an ending for both definite singular feminine (e.g., "jenta") and definite plural neuter nouns (e.g., "barna"). :P

Because et Barn is neutral

meg = me
e.g., Vil du komme med meg? (Do you want to come with me?)
mitt/min/mi = my