In Roman (and even medieval and early modern) cities the poor and working class ate food prepared off site because they lacked kitchen facilities, specifically for cooking. If you lived in an insula (apartment) or such, you would take your ingredients to a bakehouse to be cooked, or eat fully-prepared food from a taberna like this.

The wealthy would have kitchen facilities in their villa, and usually the middle class in their domus, and would probably usually eat at home (as might their servants and some clients) but even then they might have lunch on the street while in town transacting business.