Is it possible to condition oneself to not be bothered by distressing sounds?
Short answer
Meditation may be key to block out distracting noise. 

Background
One study has shown that during meditation, expert meditators with over 19,000 h of meditation experience show less brain activations in regions associated with discursive thoughts and emotions (prefrontal regions, basal ganglia, and subthalamic nuclei), and more activation in regions related to response inhibition and attention (ventral prefrontal cortex and intraparietal lobule). 

The authors used neutral (restaurant background noise), positive (cooing baby) and negative sounds (a screaming lady).

However, meditation is probably not the conditioning training you are after for use in everyday life, but since the question has been silent I thought to start off with this lead.

Reference
- Brefczynski-Lewis et al., PNAS (2007); 104(27): 11483-8