"The last thing on earth I feel physically, emotionally or mentally equipped to do, is to drive to the next city right now."
However, driving is a bit odd for this context. More likely would be: "The last thing on earth I feel physicially, emotionally or mentally equipped to do, is to deal with that troublesome relative right now."
The idea is one of psychological exhaustion. To be unable to mentally cope with stress at the time in question.
Hope this helps!
- For Teachers