Background: This pilot study aimed to investigate quality of life, psychological burden, unmet needs, and care satisfaction in family caregivers of advanced cancer patients (FCs) during specialized inpatient palliative care (SIPC) and to test feasibility and acceptance of the questionnaire survey. Methods: During a period of 12 weeks, FCs were recruited consecutively within 72 h after the patient’s admission. They completed validated scales on several outcomes: quality of life (SF-8), distress (DT), anxiety (GAD-7), depression (PHQ-9), supportive needs (FIN), palliative care outcome (POS), and satisfaction with care (FAMCARE-2). We used non-parametric tests, t-tests and correlation analyses to address our research questions. Results: FCs showed high study commitment: 74 FCs were asked to participate whereof 54 (73%) agreed and 51 (69%) returned the questionnaire. Except for “bodily pain”, FCs’ quality of life (SF-8) was impaired in all subscales. Most FCs (96%) reported clinically significant own distress (DT), with sadness, sorrows and exhaustion being the most distressing problems (80–83%). Moderate to severe anxiety (GAD-7) and depression (PHQ-9) were prevalent in 43% and 41% of FCs, respectively. FCs scored a mean number of 16.3 of 20 needs (FIN) as very or extremely important (SD 3.3), 20% of needs were unmet in >50% of FCs. The mean POS score assessed by FCs was 16.6 (SD 5.0) and satisfaction (FAMCARE-2) was high (73.4; SD 8.3). Conclusions: This pilot study demonstrated feasibility of the questionnaire survey and showed relevant psychosocial burden and unmet needs in FCs during SIPC. However, FCs’ satisfaction with SIPC seemed to be high. A current multicenter study evaluates these findings longitudinally in a large cohort of FCs.