Learn Russian: Обеим сёстрам нужна гармония.

Обеим is the dative case of обе, the feminine form of 'both'. It is feminine to match сёстрам (sisters). If you were talking about two brothers, or a mixed-gender group, you would use the masculine/neuter dative form обоим.

Why are 'both sisters' in the dative case (Обеим сёстрам)?

In Russian, the 'need' construction uses the dative case for the person who needs something. You are literally saying 'To both sisters, harmony is necessary.'

Why is it нужна (singular) when there are two sisters?

The word нужна (a short adjective meaning 'necessary') agrees with the grammatical subject of the sentence. Because of the dative construction, the subject isn't the sisters—it is гармония (harmony). Since гармония is feminine singular, we use нужна.

Why does сёстрам have a ё when the singular сестра just has an е?

This is due to a stress shift. In the singular сестра, the stress falls on the final -а. In the plural, the stress shifts to the first syllable. When a stressed е sounds like 'yo', it is written as ё, giving us plural forms like nominative сёстры and dative сёстрам.