The hospital doctors don't go out much because all their time is taken up by work.

Literal

Hospital [possessive-の] doctors [topic-は] their time [subject-が] all work [by-で] occupied [because-から] not-much go-out-not.

時間がすべて仕事でふさがっている describes time being 'blocked/occupied' by work. から gives the reason. あまり外出しない uses the あまり...ない pattern for 'not much.'