The words archaic and ancient are synonyms, but do differ in nuance. Specifically, archaic implies having the character or characteristics of a much earlier time.

// the play used archaic language to convey a sense of period

Although the words antique and ancient have much in common, antique applies to what has come down from a former or ancient time.

// collected antique Chippendale furniture

The words old and ancient can be used in similar contexts, but old may apply to either actual or merely relative length of existence.

// old houses

// an old sweater of mine