Having savings you can rely on in an emergency is a useful thing.
Literal
By-chance's time [at-に] can-rely savings [object-を] have thing [topic-は] useful thing [copula-です].
頼れる is the potential form of 頼る ('to rely on') — 'can rely on.' Used attributively to modify 貯金: 'savings you can rely on.' 役に立つ ('to be useful,' literally 'to stand in a role') is the idiomatic expression for usefulness. Note the layered noun-phrase structure with こと.