When she went to Oshima by ship, she got seasick and threw up everything she'd eaten.

Literal

She [topic-は] ship [by-で] Oshima [to-に] went [when-とき], seasickness [to-に] caught [and-て] eaten things [object-を] all threw-up-ended-up.

A rich sentence chaining multiple grammar points. ~とき marks the temporal clause ('when she went'). 船酔いにかかる uses かかる in the sense of 'to catch' or 'to come down with' — the same verb used for catching colds (風邪にかかる) and other ailments. もどす literally means 'to return' but here is a common euphemism for 'to vomit.' The ~てしまった ending adds a sense of regrettable completion — the vomiting happened and couldn't be helped. 大島 (おおしま) is a common Japanese island name, likely referring to Izu Oshima off the coast of Tokyo.