The big rusty thing is the shaft coupling which joins with the transmission output coupling. It can be removed but you will need a gear puller and the engine needs to be moved forward to gain enough space to work on it. Sometimes you can separate the two flanges and put a nut between the shaft and the transmission. Then retighten the bolts and push the shaft out that way. I would rather move the engine and work like a gentleman.
To rebuild the stuffingbox you will need to haul the boat. You could repack it while still in the water but you really need to clean everything up so wait until you are out for winter storage.
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All the best,