It is unarguable and inarguable that having more than one prefix with the same meaning is a problem. Which is the correct one to use? In the case of unarguable and inarguable, apparently both! Some combinations are not used in English and would not be understood. Sometimes the choice gives us words with different meanings.
The word unfamous, as the opposite of famous, means that something or someone is ordinary and not known to many people. However the word infamous is not exactly the opposite of famous, it's a special case of famous, but has a very different meaning. The word infamous describes a person that is famous in a bad way. The word fame is usually positive, but you can have a famous disaster, shipwreck etc.
However the un- prefix does compete for use against the Latin in- prefix both meaning not. For example we have both unarguable and inarguable. We have the word (adjective) uncivil and the word (noun) incivility.
The un- and in- prefixes do not always have the same meaning. For example: the word pair unformed and informed have the same origin, but very different meanings: unformed is not formed; informed is to have the knowledge formed in oneself.
New words created with these prefixes might not be easily recognised. In such cases the prefix might get added with a hyphen. For example the word un-American, describes an American who opposes traditional American values. Also there is in-patient meaning a patient who stays in the hospital for treatment, as compared to out-patient. The hyphen is important, to separate it from the word impatient which has an unhyphenated prefix.
These prefixes can be doubled up when the English un- gets added to French/Latin in- words used in English. An example of this is the word uninhabited. It is from the Latin inhabitare, which is in- (in) plus habitare, meaning to reside. Another example is uninfected. It is from the Latin infectus, meaning to spoil, stain or corrupt. It is formed by in- (in) plus facere, meaning to make or perform.
Of course the reverse does not happen, the Latin prefix in- does not get added to English un- words. English borrowed many words from Latin, but Latin didn't borrow from English.
If you are confused by the choice of in- or un-, it is not your failing but a very confusing point in the mess that is English.