The phrase 'that without which one cannot do' perfectly defines the word indispensable. An indispensable item or person is absolutely necessary or essential, meaning one cannot function or succeed without it. For example, a car's engine is indispensable for its operation.
Let's contrast this with the distractors. Infallible means incapable of making mistakes or being wrong, often used to describe a person or a system that is always correct. The Pope, for instance, is considered infallible in matters of doctrine. Inadmissible refers to something that cannot be allowed or accepted, particularly in a legal context, such as inadmissible evidence in court. Neither 'infallible' nor 'inadmissible' conveys the meaning of absolute necessity. Therefore, indispensable is the only word that accurately matches the given definition.