It's a good thing (that)I'm far away from my family.
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Can we omit 'that'? Is 'It' a false subject and is 'that I'm...' a real subject?
Yes, you can omit "that." With or without that word, the sentence will be fine.
You are correct about the subjects. Remember, though, that "I" (in "I'm") is the subject of the subordinate clause, not the main clause.