Java converter from Arabic to Roman Numerals
private final Map&lt;Integer, String&gt; BASE_VALUES_MAP


This is missing static modifier

private StringBuilder romanNumeralString;


I don't think that should be a member, just pass it as argument where necessary. That way you code will be thread-safe, without any additional effort.

private int findNearestBaseValue(int arabicNumberToConvert) {


If you use TreeSet instead of HashSet you could just use ceiling()/floor() instead of manual iteration.

buildRomanNumeralString()


This method looks unnecessary, just inline it.


  Are there Java 8 features that I should be using but not using.


I would say it is the opposite - there is no real point in using Stream in repeatBaseValue(), normal for loop would be fine.