Close enough to go directly to a target.
In the Late Middle Ages, when 'point-blank' was coined, archery and artillery targets were usually white. 'Blank' derives from the French 'blanc', which of course means white. 'Point' is a little more ambiguous. What was first meant by 'point-blank range' was rather more precise than our current meaning. Then, as now, it meant 'too close to miss', but the specific meaning was 'within the distance that a missile travels in a direct line, with no perceptible drop due to gravity'. The 'point' in the term may have referred to the point of the arrow that was about to be fired - if the point coincided with the target in the archer's eyeline then the target would be hit, so long as it was within 'point-blank range'. Another interpretation is that 'point' was a verb and that 'point-blank' just meant 'pointing at the (blanc) target'.
The expression betrays its ye-olde origins by appearing first in print in the form of 'poynt blancke'. An example of that comes from the English mathematician Thomas Digges' Arithmetical Military Treatise, 1579, asking the kind of question that might have turned up in a Tudor maths lesson (no wonder they crept like snails, unwilling to school):
If a Falcon that carrieth poynt blancke 150 pase, at vtmost randon randge 1300 pases, I demaunde howe farre a Culuering at his vtmost randon will reach, that at poynt blancke, or leuell, rangeth 250 pase.
Note: Falcons and culverins were cannons, small and large respectively.
Into the 17th century and the 'direct level flight' meaning was alluded to by no less an author than Sir Walter Raleigh, in The History of the World, 1614:
Training his Archers to shoot compasse, who had bin accustomed to the point blanke.
Note: Compass meant 'curved', as in the flight of an arrow over a long distance.
The figurative 'direct and blunt' meaning that we now often use is found in phrases like 'asked/denied/refused pointblank'. This also came into use around the turn of the 17th century and was listed in John Florio's English/Italian dictionary A worlde of wordes, 1598, in which he equates 'forthright' with 'point blanke'.