The growing popularity of speech interfaces goes hand in hand with the creation of synthetic voices that sound ever more human. Previous research has been inconclusive about whether anthropomorphic design features of machines are more likely to be associated with positive user responses or, conversely, with uncanny experiences. To avoid detrimental effects of synthetic voice design, it is therefore crucial to explore what level of human realism human interactors prefer and whether their evaluations may vary across different domains of application. In a randomized laboratory experiment, 165 participants listened to one of five female-sounding robot voices, each with a different degree of human realism. We assessed how much participants anthropomorphized the voice (by subjective human-likeness ratings, a name-giving task and an imagination task), how pleasant and how eerie they found it, and to what extent they would accept its use in various domains. Additionally, participants completed Big Five personality measures and a tolerance of ambiguity scale. Our results indicate a positive relationship between human-likeness and user acceptance, with the most realistic sounding voice scoring highest in pleasantness and lowest in eeriness. Participants were also more likely to assign real human names to the voice (e.g., “Julia” instead of “T380”) if it sounded more realistic. In terms of application context, participants overall indicated lower acceptance of the use of speech interfaces in social domains (care, companionship) than in others (e.g., information & navigation), though the most human-like voice was rated significantly more acceptable in social applications than the remaining four. While most personality factors did not prove influential, openness to experience was found to moderate the relationship between voice type and user acceptance such that individuals with higher openness scores rated the most human-like voice even more positively. Study results are discussed in the light of the presented theory and in relation to open research questions in the field of synthetic voice design.