The 16 is a short action 308, 243, 22-250 etc, the 116 is the long action 270, 30-06, 7mm Mag, 300 win mag etc.
There is no way to go from a short action to a long action, however, you can build a Savage 308 or 243 on a long action (30-06 sized) receiver. The cost involved can be equal to a new rifle but usually what you get for the added expense is a better barrel. You can purchase a new CBI barrel for $280 which is a entry level brand of Kreiger and still have a very good shooter. I purchased a mdl 16 from Cabela's and switched it to a Shilen 22-250 and it's an amazingly accurate rifle. I also have a Stevens 200 which is a long action in 7mm Mag, I will be converting that to a CBI barrel in 243Win with an 8 twist for 600 - 1000 yard stuff.
The nice thing about Savage is the ability to change out bolt heads and the ease of re-barreling the action, no gunsmith is required just a couple of very easy to use tools.