Learn Spanish (Latin America): Me preguntó si el rechazo me espantaba. - She asked me if the rejection scared me away.

This is an example of backshifting in reported speech. The original direct question was likely in the present tense: ¿Te espanta el rechazo? (Does rejection scare you?). Because the reporting verb preguntó is in the past, the present tense from the original question steps back into the past, becoming the imperfect espantaba.

Why doesn't the word si have an accent mark?

Without an accent mark, si means 'if' or 'whether', which is exactly what you need to connect a reported yes/no question. With an accent mark, sí means 'yes'.

Could I use asustaba instead of espantaba here?

You could, but espantar carries a helpful nuance here. While asustar simply means to scare or frighten, espantar often implies scaring someone away or driving them off. This perfectly captures the idea of being deterred by rejection.

What are the roles of the two me pronouns in this sentence?