My father can't get through even a single day without coffee.
Literal
Father [topic-は] mere 1 day [even-でも] coffee without [contrast-は] cannot-manage.
たった emphasizes 'mere/only.' ~なしでは sets up the condition: 'without X (contrastively).' やっていけない means 'can't get by/manage' — from やっていく ('to get along'), a useful survival phrase.