Problem 699

12. Except for 995, all numbers $1,2,3, \cdots, 1989$ can be divided into 994 pairs: $(1,1989),(2,1988), \cdots,(994,996)$. Since the parity of the two numbers in each pair is the same, the result of the operation is always even regardless of how the “+” or “-” signs are placed before each pair. Since 995 is odd, the sum of the numbers $1,2,3, \cdots, 1989$ with any placement of “+” or “-” signs is always odd. Therefore, the smallest non-negative number that can be obtained is not less than 1.
On the other hand, the number 1 can be obtained in the following way:
$$1=1+(2-3-4+5)+(6-7-8+9)+\cdots+(1986-1987-1988+1989) \text {. }$$

Therefore, the smallest non-negative number that can be obtained from the sum is 1.