Remember that velocity and acceleration are vectors, they have a magnitude and direction. Speed, on the other hand, is a scalar quantity which only has magnitude.
If you are moving (in a car) towards the right and apply the breaks, you are decelerating (slowing down)--your acceleration is directed towards the left while your velocity ("speed" in the diagram) is pointed towards the right (and is getting smaller):
So yes, the negative sign indicates a slowing down. A negative velocity would mean that the object is now moving in the opposite direction it was initially (or what you call "positive" direction, we could have made left the positive direction in the previous example).
The magnitude of your (negative) acceleration seems pretty high (though reasonable order of magnitude as to what I get below) given the high rate of speed initially ($v_1=150\,\rm m/s$), the zero final speed, and the extremely short stopping distance ($d=0.5\,\rm m$). You can use one of your kinematics equations,
to get that $a=-22,500\,\rm m/s^2$, which is about 1/4 of what you have. Not sure where your mistake is, but I suspect it's in the calculation of the time (which doesn't seem necessary to me).