Solution:

Work modulo $1000.$ Every factor from the third one on ends in at least three $9$s, so each is $\equiv -1 \pmod{1000}.$ There are $999$ factors in all, hence $997$ of them are $\equiv -1.$

The product is therefore $$\equiv 9 \cdot 99 \cdot (-1)^{997} \equiv -891 \equiv 109 \pmod{1000},$$ so the remainder is $109.$