To be 'off one's base' means that person is crazy or behaving in idiotic ways.
This phrase is believed to have originated with baseball, where base runners could be deemed crazy for leaning too far off the base plate, risking the chance of being picked off. The term looks to seperate itself from the sport of baseball around the end of the 19th century.
The Piqua Miami Helmet, printed in 1880, uses the expression in the 'crazy' sense, when a man is criticised for spending his money in wildly irresponsible ways:
"Yet today he is almost peniless. I saw him last night on his semi-weekly spree. As he said 'the old man was off his base again.' I have seen him bet $50 on 'faro' when so drunk he could not see the cards."
* While I was at the gym working out and trying to get back into shape, I noticed someone that had to be off his base because he was attempting to lift weights over 200 kgs!
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Note: The origins for most common idioms cannot be said with a certainty. What's provided are theories that may be plausible to how a phrase originated, but not necessarily so.
In addition, quotes that contain a phrase may be taken from old newspapers, poems, or books that were written centuries ago, but this by no means shows they originated from these. In all likelihood, if an expression is being used in a newspaper, it's probably already well known, and thus, from an older period of time.