Solid characters doesn't mean much, computers aren't that smart yet :).
You likely want to implement collision detection. Each character would have self collision detection and collision detection against other characters. You'd also create procedures to stop body parts from going further when a collision is detected. Unfortunately, this isn't generally something you can just "turn on". It'll take a good amount of work. I won't go into collision detection, since you didn't ask about it.
The easiest solution, as Nicol Bolas said, is to create your animations so that characters don't collide with themselves. Additionally, if you have characters animating next to each other, you'll have to ensure they are positioned to not collide with each other.
Here are a few things you can do to help ensure two animations don't collide:
- Calculate a bounding box for the maximum volume an animation takes place in, make sure characters aren't within each others volumes when they start those animations.
- Create a series of bounding boxes around the limbs and torso that describe the max volume of animations, again, ensure characters aren't in each others boxes. (This method will allow characters to be closer in many situations, but will require more checks.)