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I have a list of several entities, all of each have a numerical relationship to each other that defines an abstract distance.

Is there a mathematical way to plot all of these on a 2D space, turning the abstract distance into a 2D euclidian distance, and preserving all of their distances?

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    You might be asking whether every finite metric space can be isometrically embedded in the plane. According to Wikipedia (with 2 references given), it is not even possible in general to isometrically embed a finite metric space into Eucidean space of any dimension: http://en.wikipedia.org/wiki/Metric_space#Examples_of_metric_spaces2010-12-13

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