Does the category of $O_X$-Modules have exact products

Question

I was wondering if the category of $O_{X}$-Modules over a scheme $X$ has exact products. I've searched on the internet and found that it is a Grothendieck category (something that i am not familiar with but I figured that this doesn't imply the exactness of the direct products).
Also, I would like to know, if this is not the case, whether the category of quasi-coherent $O_{X}$-Modules has exact products. Since in this post it is clear that this is not always true, I claim that $X$ should be quasi-compact,quasi-separated but I can't prove anything and I am stuck. Any help or recommended reference would be very helpful.