Can a particle have momentum without energy? Can a particle have linear momentum if the total energy of the particle is zero? Even if a particle has a certain velocity, can its potential energy cancel out the kinetic energy as to add to zero ? 
 A: As weird as it sounds, the answer is "yes."
Take, for instance, a satellite in gravitational orbit around some heavy body. It's energy is given by
$$
H=\frac{p^2}{2m}-\frac{GMm}{r}
$$
Clearly, there are solutions to this equation which have $0$ energy (look at a slowly moving particle that's really far away), but those solutions necessarily involve a non-zero momentum.
This answer may seem artificial because it also allows for negative energies (oh! horror of horrors), but mechanically, this gives all of the correct equations of motions. The details are very enlightening, too: positive energy orbits correspond to hyperbolae (unbound orbits, scattering orbits), negative energies correspond to ellipses (bound orbits like those described by Kepler's Laws or, colloquially, just "orbits"), and zero energies correspond to parabolae. Parabolic orbits are non-periodic, but never escape the effective gravity of the heavy mass (unlike hyperbolic orbits). These shapes, of course, all collapse into lines when the particle has zero angular momentum.
Edit: As David mentions, in relativity a free particle with zero energy simply does not make sense because it would have to be both massless and without momentum. Massless particles are massless in all reference frames, so the particle would have to be momentum-less in all reference frames as well (which sounds like a pretty boring particle to me). But if you include interaction potentials in your definition of a particle's energy, positive, negative, and zero energies are possible once again.
A: As I cannot post any comments I have to post this as an answer although the essential points were already given:
In classical mechanics energy itself was no meaning. Only energy differences have a physical interpretation. Thus in the classical case energy is only defined up to an arbitrary constant. So any fixed state's energy can be set to zero (but of course not all state's at the same time).
As correctly stated by David, in special relativity, this constant has to be chosen in a particular way (As energy there is directly connected to mass. Mass of course also has a meaning without looking at mass differences).
So the answer really depends on the context here.
A: yes a body can have momentum even its total mechanical energy is zero as all energy are relative quantity so if a body is not bound to any system then momentum can be exist if observer is present on that body itself which is not bound to the conservative force
