'Me' indicates that it is, in fact, I who was on the run. An example:
Busting a quickie all over my belly right before getting out of bed is one of the very few things that keeps me going. (A universal sentiment)
HIM busting a quickie all over my belly right before getting out of bed is one of the very few things that keeps me going. (Straight up homosexual)

可以这么理解，但look like后面一般直接接从句，不用that，用法很奇怪，但口语使用广泛，不知道书面语妥不妥。
宾语从句的主语是me being on the run for 5 days，这里me应该也是口语的用法，用my应该更恰当。动名词(事件)的逻辑主语需要明确时写出。常见的：
Do you mind me/my smoking here?
你介意我在这抽烟么？
mind的是my smoking here这件事，而不是me这个人。

印象中你研究语法已经有相当一段时间了，效果看来不太好，有没有考虑通过扣语法来学英语的方式不太适合你？

Sup bro , I dont speak f k ing english man. you know what im saying?

等于my