Background: Vascular strokes are the leading cause of long-term disability for adults. They impose high levels of burden on the patient, the family, and national healthcare systems worldwide. This study aimed to assess the effects of patients’ and caregivers’ characteristics on the perceptions of burden in families caring for a loved one living with stroke in Greece. Methods: Using purposive sampling, 109 dyads of patients and their respective caregivers were recruited from the Attica region. Patients completed a questionnaire that included personal characteristics and the Barthel Index, while caregivers completed a set of questionnaires—personal characteristics, revised Bakas Caregiving Outcomes Scale (BCOS), Personal Resource Questionnaire (PRQ 2000), and Center for Epidemiological Studies-Depression (CES-D). Results: Caregiving burden was linked to both patients’ and caregivers’ characteristics. A patient’s educational level, the number of family members living in the same house, the existence of equipment and facilities in the house, and the duration of provided care were associated with perception of greater burden. Regarding caregivers’ characteristics, those in good health had a significantly lower perception of burden. Higher PRQ 2000 scores were significantly associated with higher BCOS scores (less burden), and higher CES-D scores were significantly associated with lower BCOS scores (more burden). Conclusion: Caring for a loved one affected by stroke places a considerable burden on the caregiver. Systematic assessment and intervention strategies can help to identify caregivers at risk so that suitably targeted assistance may be provided.