The meaning of this term does not come from a simple combination of these characters. Instead, it is a direct quote from a historical event. The phrase was a notoriously ambiguous reply, which can be interpreted as “Perhaps there is (evidence)” or “It is not necessary that there is (evidence).” The villain in the story used this ambiguity to justify an execution, and the phrase has since been immortalized to mean a charge that is completely fabricated.
The entire meaning of `莫须有` is rooted in one of the most famous and tragic stories in Chinese history: the tale of General Yue Fei (岳飞). During the Southern Song Dynasty (12th century), Yue Fei was a brilliant and patriotic general who was on the verge of defeating the invading Jurchen armies and reclaiming lost territory. However, a treacherous and corrupt chancellor at court, Qin Hui (秦桧), wanted to make peace with the invaders. Fearing Yue Fei's success and influence, Qin Hui recalled him from the front lines and had him imprisoned on charges of treason. When another general, Han Shizhong, confronted Qin Hui and demanded to know what evidence he had of Yue Fei's treason, Qin Hui infamously replied with just three words: “莫须有 (mò xū yǒu).” This vague, non-committal answer was all that was given to justify the execution of China's greatest general. Yue Fei and his son were subsequently executed. The term `莫须有` became a permanent cultural shorthand for the ultimate abuse of power and a miscarriage of justice.
`莫须有` is a formal, literary, and very strong term. It is not used for trivial, everyday disagreements. You wouldn't accuse someone of `莫须有` for wrongly thinking you took the last cookie.