Why are all the spearman's rho correlations with one variable low when the other variables all have moderate correlations? I was wondering what could be the reasons for a very low Spearman correlation, as low as 0.01-0.06. Basically no correlation even exists!
All my other variables have moderate correlations except any correlations with this one particular variable.
 A: Spearman's correlation coefficient allows to assess if there is a monotonic relationship between two variables. 
A very low coefficient means that the relationship is not monotonic. There might be no relationship at all, or there might be a more complicated non-monotonic  relationship. 
The easiest way to understand what is going on is to look at scatterplots. They will tell you much more about the relationship between two variables than a correlation coefficient can do. 
A: @lejohn has provided a clear answer. I just wanted to provide a few observations from my experience as a statistical consultant in psychology. All the following observations pertain equally to Pearson's and to Spearman's correlation.
Of course it is not surprising that you can have a set of variables that are all intercorrelated and a focal variable that isn't correlated with anything else.
What your question implies is that this pattern of results is not as you were expecting. Thus, in addition to understanding the results, it is worth reflecting on where your expectations came from and how you should update your understanding of the phenomena in light of the results.
In psychology, I find that the scenario you describe happens often when researchers think too much at the level of constructs as opposed to the level of measurement. For example, you might have two measures of human performance where one is based on an objective measure of performance and another is based on self-report. At the level of constructs, they are both measures of the same thing, and thus should be correlated, but at the level of measurement the correlation may be weak to non-existent.
Thus, if you mix measurement types (objective, with self-report, with other-report, etc.; quantitative with qualitative, etc.) often correlations can be substantially lower than what happens when you work within a measurement type. Thus, your expectations may be informed by experience reading papers that have used a common measurement type; as a general rule, in psychology it is important to think deeply about what is actually measured rather than thinking purely about the concept of what is meant to be measured.
