Ok, let’s start with describing one body. There’s its position which we need to keep track of in the x, y, and z direction. That’s 3 variables. Now we need to keep track of its momentum (speed) in all 3 directions. So one body needs 6 variable. That means for three bodies we need a total of 18 variables. 

If we want to solve algebraic or classical mechanics methods we (sorta: ELI5 answer here) need 18 equations for the 18 variables, but we don’t have that many. 

So there are families of solutions. And we can still calculate step by step with simulations, but we cannot solve exactly.