What do the words 'tedious' and 'terminable' mean in the sentence?
At that moment I was reminded of one of the most tedious of his seemingly terminable anecdotes.

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"Tedious" means boring. "Terminable" means finite. In this sentence, don't you mean "interminable?" "Interminable" means "never ending".

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'tedious' means the anecdote (story) was boring.
'terminable' means, 'able to come to an end.'
By 'seemingly terminable anecdote' the author implies that his stories are generally long-winded.