Classical state of a particle lives in the 2n-dimensional manifold known as phase space. The coordinates of the phase space are the configuration space and the corresponding conjugate momenta. The state is a single point on the phase space, corresponding to the initial conditions of the problem. The Hamiltonian determines the time evolution in the phase space, which is a trajectory.
To make is simpler for you to understand: 1. Position and momentum couples form the physical state. And of course, for any physical system, it exists. 2. You need 2 initial conditions to solve a 2nd order differential equation. You can't solve a differential equation without initial conditions (of course to use this fact, you would need to understand that F=ma is a differential equation, not a function) (Maybe it's better if you read some good introductory text, such as Feynman Lectures on Physics Vol 1, Chapter 9).
If still not clear for you: the initial state in a two body problem is determined by the initial position and velocity. They are the initial conditions of your "function" F=ma, which is actually a second order differential equation. F determines how this initial state will change over time.