Answer: 109

Small Hint:

Every factor from the third one on ends in at least three $9$s, so it is $\equiv -1 \pmod{1000}$

Big Hint:

Count how many factors are $\equiv -1 \pmod{1000}$ and check whether that count is odd or even; only $9 \cdot 99$ remains to compute

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 $$\begin{aligned} &\equiv 9 \cdot 99 \cdot (-1)^{997} \\ &\equiv -891 \equiv 109 \pmod{1000}, \end{aligned}$$ so the remainder is $109.$