The simplest definition of the idiom “a month of Sundays” is “a very long time,” though like many sayings, it’s possible to dissect this expression and find more literal meanings and cultural origins. For instance, a person might reference the literal idea of a month filled with Sundays, which would reference the time it takes for 30 or 31 Sundays to pass. He might use the saying to refer, directly or indirectly, to the religious and cultural connotations of having a month filled with Sundays or a time period of limited or unexciting activity. Some people use the saying when referring to an event that is impossible or unlikely to happen, just as a month will never be filled with only Sundays. Still, although it might not be universal, this idiomatic expression is widely accepted among many English-speaking cultures as one that means a particular event or time period is extremely long.
Taking the idiom “a month of Sundays” at its most literal sense, it describes a period of time that is longer than a month. For example, most months have 30 or 31 days in them, and during those months Sundays come along every seven days. So, to think about a month filled with Sundays is to think about 30 or 31 Sundays coming along every seven days. This would take 30 weeks, or about seven-and-a-half months. In this sense, one friend might complain to another that she hasn’t seen her in a month of Sundays, meaning the two haven’t seen each other in a long time.
When referencing the religious or cultural aspects associated with Sundays, someone who says an event or a time period has lasted a month of Sundays might refer to the atmosphere of that time. For example, people of various religions and cultural beliefs have long designated Sundays as days of worship and rest, and some cultures ban people from engaging in any kind of entertainment on Sundays. That inactivity can lead to boredom or weariness, feelings that can make time feel longer than it is. So, to have a month filled with Sundays can be to have 30 or 31 extra long, and perhaps boring, days.
Given that a month of Sundays is impossible, some people use the saying to describe an event that will never happen. Perhaps a football coach says his third-string quarterback will earn an athletic college scholarship in a month of Sundays. When used in such situations, usually the saying means the event won’t occur.