I wonder if there is any way to travel to an island (in my case: Ireland) without taking a plane or a boat.
My sister claims to be frightened of both, so is there a way for her to get there?
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Ireland is an island and there are no tunnels or bridges (or any other means) that would be suitable for cars, buses or trains.
Sorry, the only practical way to get to Ireland is a ferry or airplane ride. Both are relatively short: the plane ride is about an hour and the fastest ferry from Holyhead to Dublin takes about two hours.
You will have to choose between visiting this beautiful island and some travel inconvenience. Keep in mind; it's only a relatively short trip (from England) and many thousands of people do travel to Ireland every single day without any problems.
If she won't fly (I assume a chartered helicopter is as out of the question as fixed-wing aircraft; they're more dangerous), and she won't ride in a boat, there is only one way left, and I intend this answer with all seriousness, not as a joke: swimming.
The water is too deep to permit wading, and there are no bridges between Great Britain and the island of Ireland. (One can get to the island of Great Britain by train from the mainland of Europe, but this has only been possible since the 1990s.)
She simply won't be able to go unless and until a bridge is built, and while this is a technical possibility, it is not on any government's radar at the moment and these projects tend to take many years to fund, design and build. It seems unlikely that a trip to Ireland by land transport will be possible until at least the 2030s.
These people have swum the North Channel between Ireland and Great Britain, but they seem all to have been swimming specialists in extraordinary physical condition, so it is not a journey for the faint of heart - and assuming she doesn't intend to live out her days in Ireland, it would require a return journey via the same method.
If parachuting is allowed, there are two options: a human cannonball (although the world record distance is only about 200 feet or about 60 metres; or by catapult, when an altitude of 300 feet or 90 metres has been achieved.
But more seriously, the most feasible means of transport are by airship or hot air balloon.
There are several companies offering chartered helicopter trips, e.g. https://www.atlashelicopters.co.uk
An initial quote from them for flying London -> Dublin is about 7000 GBP, for two passengers. The price is heavily dependent on the time, and this particular stretch would be roughly ~2 hours. You can cut down on costs by finding a helicopter company closer to Ireland.
There are also many Hovercraft companies both in Ireland and England that could possibly allow chartering a trip between the islands, however for traveling over the open seas a large high capacity hovercraft would be required, and as there are no active routes catering to many persons, this would be very expensive. There used to be a hovercraft route to Isle of Man however, but this has been replaced by catamarans. The only active hovercraft route in the vicinity is by Hovertravel and goes between Portsmouth and Isle of Wight https://www.hovertravel.co.uk/
For getting to England, you can go by tunnel from mainland Europe (train, car, bus...).
Rent, buy or hire a hovercraft ferry.
There are some services in the UK that offer hovercraft vehicles.
One such company is (linked just as an illustration, no endorsement implied):
Obviously, it is possible to swim.
The question doesn't explain why boats cause fright. If it's about being in control of the circumstances, then water skiing may offer a feeling that being on a boat doesn't.
Crossings between the UK mainland and Ireland are 12 to 50 miles long. Such a crossing would be feasible given the right equipment and a moderate amount of experience.
[Thanks to PLL for the suggestion.]
Perhaps if you get lucky, you could board a passing by iceberg that may be floating by caused by global warming (if you believe such things). Icebergs can be many miles in diameter and would not likely cause sea sickness. You will have to construct some sort of apparatus to drive it over to Ireland though. Or if you two are patient, you could wait a few hundred million years and hope the tectonic plates will eventually connect you to Ireland. Maybe there will be a bridge before then though if you can't manage to live that long.