I think it's not possible
Ferries normally ply their trade over relatively short crossings. The shortest crossings between Afro-Eurasia and America would either be between Siberia and Alaska or From Greenland to Canada, possibly via Iceland. You would need both to make a "round the world" trip.
As far as I can tell, there are no ferries from Greenland to either Iceland or Canada.
Crossing the open ocean wouldn't be a "ferry" any-more. It would need to be a much larger ship. In the days before airliners there used to be ocean liners plying their trade across the worlds oceans but airliners basically killed them off, afaict there is only one left and even that is more of a toy for rich people than a practical mode of transport.
How close can you get?
Assuming you can navigate any bureaucratic issues You can start on the West Coast of Iceland. Drive across Iceland, take the ferry from Seyðisfjörður to Hirtshals, Denmark and then travel through Eurasia until the road runs out somewhere in Siberia.
The east coast of Iceland is at around 24 degrees west. The furthest east numbered road I could find in Siberia was p-504 to Magadan which has it's eastmost point at 152 degrees east. However, there seems to be a minor road branching off at "Zero. Povort on Omsukchan" ending at 155 degrees east. It's quite possible that there are other roads going further east. So you can get about half way round the world.
Alternatively, after crossing from Iceland to Denmark you could head southeast. If you can negotiate the bureaucracy of the different countries along your route, then there are roads all the way to Malaysia. It's not 100% clear to me if it's possible to reach Australia by ferry. Even if you do manage to make it to Australia though, the Eastern tip of Australia also seems to be at about 152 degrees east.
I don't think you can beat that in terms of furthest west to furthest east by road and ferry. I already mentioned the frozen north and there also doesn't seem to be any ferry from Australia to New Zealand. As has been pointed out in the comments if you instead measure by great circle distance the result may be a bit different.
Technically, the Cunard connection from Southampton (UK) to New York is not a ferry but a cruise, but since it goes back and forth, I would count this one as a ferry, and you can technically ( probably super expensive) take your car with you. So the Atlantic Ocean can be crossed.
When I searched this I found that the Queen Elizabeth 2 had the ability to carry cars in the past, but it seems the service was discontinued some time before the ship was withdrawn and its replacement, the Queen Mary 2, has no such provision.
would it be possible to jump on a cargo ship in a busy harbor?
Freighter travel was certainly possible pre-covid, but I'm pretty sure it was something that had to be arranged in advance. Freighter travel was suspended as part of measures surrounding the Covid pandemic. It looks like some lines may be reopening but it's still early days.