When someone "took up housekeeping" it means they started managing a household. Some synonyms for this phrase include "established a home," "began homemaking," "started running a household," and "initiated domestic duties." Other phrases that convey the same meaning include "assumed household responsibilities," "commenced house management," "embarked on domestic work," and "set up house." No matter which phrase is used, it alludes to the start of a new chapter in life, one that involves taking care of a home and the people who inhabit it.