Exeter Hospital | 7,039 followers on LinkedIn. Exeter Hospital, part of Beth Israel Lahey Health, is a 100-bed tax exempt, community-based hospital and one of three affiliates of Exeter Health Resources. At each of the affiliated companies, we are committed to providing health care services that are innovative, progressive, and focused on quality and the well-being of patients. The hospital started in 1896 as a small cottage hospital on Pine Street in Exeter.