There are many Stack Exchange clones, see this meta answer for a list.
Alternatively, with a bit of thought you can take a standard collaboration system such as SharePoint, Yammer, or a generic Wiki and add either internally developed or 3rd party add-ons to provide the reputation and other gamification features.
However, you will encounter some problems.
Many people use Stack Overflow passively, only as a resource without asking or answering questions. Others just ask questions and don't answer. As everyone else has said Stack Overflow works on sheer weight of numbers, you can look at some of the smaller sites in the Stack Exchange Network to see what might happen with less users.
Secondly if you're setting this up in an established company you will already have information stored. Probably in more than one system. In order to maximize the value of your new tool you'll need to move all the data into it from the old systems and disable the old systems to prevent people using them. In all but the smallest and newest companies this is a massive task and will get a lot of pushback (I know, I've been involved in projects trying to do similar). You'll have to sell the 'why' of moving to the new system and argue that it offsets the effort.
And, again as others have mentioned, you need to make sure that the effort required in moderation is really paid off in the benefits that people get from the system.
Lastly, because you need to put everything together you may run into a problem of compartmentalization of information. It may not be possible to share all information with all employees either for legal or intellectual property reasons. So then you need some sort of access control but it also further reduces the number of people who can interact with the whole system making it less viable.
It's certainly possible to do, a lot will depend on how big and old the company is and what the culture is. It's work excellently for some companies while it'll die a death in others (and potentially take your career in that company with it if it's regarded as "your idea").
Like anything I'd take small steps, try and move to a single system for all the data first, then look at adding the extra features. Maybe run a trial on a smaller set of the data and get feedback.