We probably wouldn't even realize it, because being part of the Universe, we'd stand still as well. ;-)

Update:

Ok it depends on which meaning of "energy" you refer to. In the physical sense, we'd have to keep in mind though that the giant clockwork of the universe is already in movement, so it would actually require a lot of energy to bring things to a halt or to neutralize all existing energy, which is not really possible and doesn't make much sense, because where should it go? Energy cannot be destroyed, only transformed after all.

Anyway, that aside, energy standing still basically means zero movement of particles, means everything being frozen to 0° Kelvin, the absolute zero-point, including ourselves. We'd be in complete, unconscious hibernation.

Now another interesting aspect of the question is whether we have a spirit or a soul that can leave the body in that state. Science would deny that of course. Now if you include our immaterial spirits as some kind of energy in your question, standing still too, then the answer would be that we'd again be in some kind of hibernation, this time in a more "spiritual" sense. If you mean our spirits to be a different quality than the energy in your question, then we would roam through a static, motionless, freeze-framed material world.

Everything would cease to exist, the chair you sit on is solid because of all the atoms moving around so much that we feel and can sit on it, now our bodies are as well solid because of all the atoms moving around so fast, these movements create a clash between the chair and our bodies thus the solid effect that we can sit on that chair.

Hence motion is life and existence, without motion nothing could exist.

maybe if energy stood still we would need to change/transform out of our material forms to continue to exist until energy woke up again and we could find our way back into it. from what has been said, nature replicates herself and maybe we, our earth, our sun, our solar system, our universe, and so on up/out, all take periods of rest before our next performance,
filled now with more wisdom than prior to our most recent bout of standing still
good question what or where are we when matter rests

if energy stood still usely nothing move or it becomes froozen. if energy stood still it would also be zero energy. and if it is zero energy then you always have energy because energy would be like a batterie. but can you tap in that energy?

good luck.

experience and enjoy.