What accelerate and rest at the same time?

I'm not sure if this counts, but if you're looking at two different reference frames then a lot of things can be at rest and accelerating at the same time. For example, if you're standing still on the surface of the earth, you are at rest with respect to other people around you. However, the earth is moving, so you are also accelerating with respect to the universe.

Is this in the reference frame of the Universe?

If it is, then you could say nothing is really every at rest, so the question would be trick one.

If this is in the reference frame of the earth:

Just by standing on the surface you experience the acceleration of gravity by not flying off into space. You also are at rest in the reference frame of the earth since you are not moving relative to the planet.

The planets maybe?

Things are said to accelerate when they change velocity, so something that is changing direction is accelerating whether is speed changes or not.

The planets move in circlular orbits around the Sun and are therefore constantly accelerating (i.e. constantly changing direction), however there speeds around the Sun are constant. Someone on the Earth is in the EARTH'S INERTIAL REFRENCE FRAME and therefore is "AT REST" with respect to the Earth and does not feel the acceleration when the Earth changes direction.

It is a trick question because "what" seems to refer to something inert while "rests" seems to refer to something living.

Whatever is FREE FALLING accelerates without expending any energy. Let's say you run toward a cliff, your are spending energy. Then you jump and during the free fall, you can rest !

no one thing does both at the same time, a dual nucleus based being might be able to acheive this but not a single organism & no machine or made item can