If you want someone to use their "best endeavours" in a contract, is that any different to asking for their "reasonable endeavours?" You might consider it reasonable to think so. Like many things in law, however, reasonable minds can differ, especially on the meaning of the word "reasonable".
This is an extremely high threshold of responsibility.
To use the earlier example of a contract wherein a party has to keep disclosed data secure, under this threshold, that party would be responsible for anything that went wrong with maintaining the security of that disclosed data, regardless of any steps they took to keep it secure.