Learn English – Is there one word for ‘six and half-a-dozen’

There is no such term. You could use semihemisphere, but this is not a geographical term as far as I'm aware. Evidence of absence is always difficult and doesn't make a rewarding answer.

I would suggest a different term altogether like area or region. If you insist on the "quarter of the earth" sense, then quadrant would fit your needs.

Carlo_R has an interesting point, but quadrant is in use, geographically, and is a word that most speakers will glean meaning from. Some examples include "Mapping World Heritage Sites":

Use a dry erase marker to draw, or have a volunteer draw, the Equator and prime meridian. Then label each quadrant of the map with its hemispheric identifier: NE, NW, SE, and SW.

"Geographic Coordinate System":

The origin of the graticule (0,0) is defined by where the equator and prime meridian intersect. The globe is then divided into four geographical quadrants that are based on compass bearings from the origin. North and south are above and below the equator, and west and east are to the left and right of the prime meridian.

Earth quadrant from Webster's (1913):

n. a fourth of the earth's circumference.

Since you're asking in a psychological context, are you looking for one of:

Detachment (detached)

Dissociation (dissociated)

Depersonalization (depersonalized)

Emotional numbness (numb)

Based on the updated question, perhaps you're looking for:

Nihilism (nihilistic): Nihilism is often discussed in terms of extreme skepticism and relativism; for most of the 20th century it has been associated with the belief that life is meaningless. Existential nihilism begins with the notion that the world is without meaning or purpose. Given this circumstance, existence itself–all action, suffering, and feeling–is ultimately senseless and empty.

From the Encyclopedia of Philosophy article on Nihilism, though there are plenty of other choices (e.g. Wikipedia).