As discussed on the US Rifle Teams forum

The word “bullseye” is a colloquialism from the sport of conventional pistol shooting. The term comes from the shape of the targets used in this sport as Conventional pistol events are fired on a target with concentric scoring rings. The centermost rings are blacked in, so that the target looks like a “bull’s eye” – a large black dot in the middle of the paper.

Point blank range is incorrectly used to imply a range so close to the target that one cannot miss. The original sense, however, is slightly different. It refers to the range close enough that one can aim an artillery piece directly at its target without adjusting for the fall of the shot.

But where does the term come from? The point seems obvious enough, as it is from pointing at the target. It is the blank that is confusing. Blank, in this case, is an old term for the center of a target as the centers of archery targets were frequently painted white, and blanc is French for white.

Also, the first artillery elevation charts had numbers to identify the different angles to identify them. Horizontal was on the chart, but had no identifier, not even a zero.  So … point blank.