My bathroom basin (sink) has developed a crack from the back of the plughole going up towards the overflow hole.  Is it possible fo repair?

There is no long-term repair for a crack like that. There are temporary fixes that will look horrible but that might last until you replace the basin. Sorry.

nope.  the weight of the sink and water will make the crack ever larger and even epoxy won't hold it together

Of course, whether the sink is solid vitreous, porcelain on iron, glass fibre, plastic, glass, tin, copper, concrete, etc etc etc is immaterial so no need to tell us.