Just put your text in quotes. Double quotes if you are in VB Script mode, and double or single if in Python mode.
To explain: When you just enter the numbers, you are giving Field Calculator a value in integer form, which it will convert to a string for you in order to enter it into the string type field. For integers, the leading zeros have no value and no meaning, so they are omitted. If you put it in quotes, you are instead passing it a string, which is already the correct data type, so it gets left alone.
This also assumes that this field is actually of type "Text". Maybe it was set to Long or Short Integer, that would also cause this behavior.