When a person drinks only one or two beers, then stops drinking, the person will inevitably become tired/drowsie. This phrase describes that phenomenon.
Since I've had lunch I've got the onebeer drowsies, should have drank three or four, at least then I'd be able to stay awake.
When some resigns or retires from a job, they are on "eternity leave". This can also refer to someone who has passed away, especially someone who worked somewhere for a long time, retired/resigned, then passed away later.
Oh I'm sorry, you didn't know that Kelly is on eternity leave? She resigned and moved to Hawaii!
With regrets I would like to inform you that my boss is officially on eternity leave. I will never see or hear them again. Ever.