In order to answer this question I would have to know about your nationality. In my experience, the two most common types of "overstay" situations are in regard to (1) foreigners and (2) overseas Chinese. I will discuss these in more detail below.
In the case of foreigners, if you have overstayed your visa, the Police will want you to leave. You will have to pay a fine of a few thousand NT$. You are best advised to prepare a written statement of what you did during your overstay period, where you lived, etc. Normally, after you leave you will be blacklisted for the amount of time that you overstayed your visa. However, if the Police get some evidence that you have been working illegally, they will double the period of blacklisting. In my years of experience in working with Taiwan immigration matters, I have never heard of a foreigner having his "overstay" status cancelled due to extenuating circumstances.
In the case of overseas Chinese, in general the situation is the same as for foreigners. However there is another possibility, which would be to institute legal proceedings to have the "overstay" status cancelled, and apply directly for an ID card to live in the Taiwan area long term. This is very difficult, and would probably take two years or more to complete, however I suspect that it could be done.