  Physical foundations for relativistic spacetimes are revisited, in order to
check at what extent Finsler spacetimes lie in their framework. Arguments based
on inertial observers (as in the foundations of Special Relativity and
Classical Mechanics) are shown to correspond with a double linear approximation
in the measurement of space and time. While General Relativity appears by
dropping the first linearization, Finsler spacetimes appear by dropping the
second one. The classical Ehlers-Pirani-Schild approach is carefully discussed
and shown to be compatible with the Lorentz-Finsler case.
  The precise mathematical definition of Finsler spacetime is discussed by
using the {\em space of observers}. Special care is taken in some issues such
as: the fact that a Lorentz-Finsler metric would be physically measurable only
on the causal directions for a cone structure, the implications for models of
spacetimes of some apparently innocuous hypotheses on differentiability, or the
possibilities of measurement of a varying speed of light.
