For the interested reader, i.e. the interested computer games programmer that wants to get closer to reality:
Actually this is not the main problem. Distances are large and floating point numbers have a precision of 7 digits! But when integrating the equations of motion numerically, you will probably run into problems depending on the method of integration. (here "integration" = solving over time). The most common approach in computer games is to calculate the acceleration from the force, add it to the velocity and add the velocity to the current position. This is known as the Explicit Euler Method, which typically increases the energy of a system (see image below). The reason is that the time step dt is not even close to zero, and we cannot make it arbitrary small because we would need to make more steps and hence decrease performance.
There are more advanced methods that take the change of the parameters over a time step dt better into account. For example I use a classical Runge-Kutta algorithm of order four for my flight simulator project. Though it looks exact for the Kepler problem, the energy decreases slightly. This is, because the phase space has four dimensions (two positions (x,y) and two velocities (v_x, v_y)).
I recently completed a project about numerical integration at the university, here are some results. This image shows the change of initial energy over time.
And the next image shows the orbital perturbation caused by the method of integration. The symplectic Euler method preserves energy, but there is some sort of angular pertubation, as in the Midpoint method. The naive Explicit Euler method increases energy, as explained above. The Runge Kutta (RK4) and Gauss-Legendre methods are quite precise and almost overlap.
The thing with conical functions in KSP is nice, as it is an analytic solution to a two-body problem. But what if the spaceship enters the atmosphere? In this case the Explicit Euler method fails, if you want to come close to reality.