Embodied conversational agents (ECAs) show promise for use as companions in emotional support applications, although real world uptake can be low. Incorporating closeness building techniques to interactions may improve engagement. Research has identified a range of linguistic, behavioural, emotional, and appearance strategies to build rapport and working alliance with ECAs, however substantially less research has investigated closeness; a relationship quality that is more suited to ECAs in supportive companion roles. Exploratory qualitative research may help to inform theoretical models and the design of ECAs for closeness in supportive applications. Qualitative data were collected as part of a mixed-method experimental study. As part of the study, a community sample of 197 adults provided written responses to two open-ended questions assessing barriers and facilitators to closeness and emotional support seeking with an ECA. Data were analysed by two independent raters using conventional content analysis. Parent themes and sub-themes were derived from the data and refined to achieve agreement. Factors affecting closeness with an ECA included its physical characteristics, conversational ability, rapport building behaviours, and program errors. These factors, plus features of ECA specific support and preferences for human support affected willingness to seek emotional support from the ECA. The results contribute to the theoretical framework of embodied agent-patient communication and may inform the design of companion ECAs. The findings suggest that companion ECAs should deliver rapport building behaviours and high quality conversations, be constructed with humanlike facial features, and be error free. Although, these features should be further evaluated experimentally.