With the increase in the elderly population, institution-based care has become another option for elderly people. In Changhua, Taiwan, the number of long-term institutions has doubled in the past decade, and more families are choosing to send their elders to institutions for care. However, there is stress induced by having to care for these elders when they come back to their family members when hospitalized. Therefore, this study aimed to understand the stress and coping behaviors of family members in regard to hospitalized long-term care facility residents and identify relevant factors that affect and predict the stress and coping behaviors exhibited by these family members. In this study, a quantitative and cross-sectional survey was conducted using the convenience-sampling method; family members of long-term care facility residents hospitalized in a regional hospital in central Taiwan were selected as the research participants and a total of 162 family members were admitted. The data were collected in the form of questionnaires including basic information and data on the stress and coping behaviors of the family members. The data were collected and coded by using SPSS 22.0 to perform descriptive and inferential statistical analysis. The standard average score of total stress for family members was 57.03 points, which corresponds to a moderate level. The four perceptions of stress by family members were, in order, physiological, life, psychological, and economic. Furthermore, family income, work status, and the relationship between residents of the family members and physiological, psychological, and economic factors had predictive power for their problem-oriented coping behaviors, with an explanatory power of 59.6%. Life aspects, gender, marital status, and the number of hospitalizations in half a year had significant predictive power for the family members' emotion-oriented coping behaviors, with an explanatory power of 19.0%. The family members had high levels of stress, especially physical stress, and the total scores of stress perception were higher for those who were younger than 39 years old and had no rotating family members. Additionally, the coping behavior of the main caregiver was mainly problem-oriented. The results of this study may serve as a reference that can help nursing staff in clinical or long-term care facilities to provide or develop effective and individualized services for family members of facility residents.