Islands with no cars sounds like a great idea: Koh Jum, Thailand
...and it actually can be a great thing but there are certain things you have to look out for. Some islands claim that they didn't allow cars on the island in order to look after the nature but I don't think I really buy that statement. I've been to 4 different car-free islands in the world and half of them felt as if the "no car" aspect of the island was more of a gimmick than an actual feature. The gimmick being that because of the lack of tech they are actually going to charge more for everything there.

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One of the first places I went to that didn't have any cars was Pu / Jum Island in Thailand and this place was pretty amazing if you were willing to forego a lot of luxuries that we all have probably gotten used to in a certain sense. Keeping in mind that I went there for the first time nearly 20 years ago and have only been back once about 10 years ago, I can imagine that things have likely changed.
This island was already almost completely devoted to tourism when I first got there, and the fact that you had to walk everywhere didn't really change what you were going to do very much because there wasn't really anything to walk to. All the bungalow operations were on the waterfront and they all shared the same idyllic view of the crystal sea and all places offered food in-house... which is what basically everyone did.
While this may have changed the method that we used to get there was to by a ticket to Koh Lanta from Krabi and then about halfway longtail boats would come and pick you up from the ferry and take you into your bungalow operation right up on the sand. This was no problem for me but I could see it being an issue for older people or those with mobility problems since it isn't exactly easy to get into a longtail boat from a ferry, nor is it necessarily easy to get onto the shore from a longtail boat. My parents would have struggled with this, my grandparents wouldn't have been able to do it at all.

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The island is very small and while I wouldn't relish walking from the north to the south, everyone can walk to the central town where the only market and signs of non-tourist civilization exist in the area. There isn't much reason to go to this town and most of it, even 10 years ago, had been converted to a tourist trap sort of thing with a few overpriced restaurants and a market and that is about it. There was a police station but I kind of laughed about how there was nobody there. The island, according to locals, is self-governing and I guess that is good.
One thing to consider that could be a factor for some people is that this island's residents are entirely Muslim, so getting a wide variety of cocktails and beers isn't easy. Some places don't have it at all. It's not like it is impossible to get a beer but there isn't much variety.
Also the electricity is quite unreliable on the island and up until recently they didn't have mainland electricity and relied exclusively on generators for power. This has changed for most or perhaps all of the bungalow operations there now but there was a time when the lights went out at 9 or 10 pm and then you simply didn't have power until the following day. Many of the cheaper bungalow operations continued to not feature electric in the rooms but only in the central area which was quite annoying if it was the middle of the night and you were burning up with not so much as a fan to cool you off. This is something to keep in mind if you are ever considering going there.

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Over the years luxury brands have started to open there and I was fortunate to meet the manager of Koh Jum Lodge while I was visiting and saw the luxry that they have there with a large pool that must be really difficult to maintain as well as bungalows that would make Swiss Family Robinson a bit jealous. There are also very simple bungalows made out of bamboo that when I was visiting could be acquired for $8 a day and they were definitely no frills.
Everyone shares the same beach though and due to the near complete lack of vehicles and no reason for cars since the "roads" aren't wide enough to accommodate them, the place is dead quiet all the time.

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For many people this might be exactly what they are looking for since peace and quiet seems to be so difficult to come by these days no matter where you are in the world. The only noise I heard when I was there both times would be when a longtail boat fires up and you only have to endure that for as long as it takes the drive to get about 50 meters away. Other than that this island is very quiet all day and night long unless someone is having a party and from what I have been told the parties have strict rules on noise levels. This is all fine by me.
It is pretty accessible too since the international airport isn't really that far away. The last I heard the Krabi airport was expanding yet again so I would imagine that the easy ability to get to Koh Jum has only gotten better.
As far as "no car" islands are concerned I think that Jum is one of the better ones and the only thing that I can think of that would be a negative is that I do feel as though they take the piss with prices on this island on a regular basis. They can get away with it as well because there is no competition. I don't foresee them ever selling up a ton of land to Holiday Inn either since this small island basically isn't for sale and has been owned and managed by the same families for as long as anyone can remember.
If that sounds like your thing it is really easy to find. And from my rather extensive experience island-hopping around the world, this is one of the places that I remember the most. But just don't plan on there being a world of activities available to you here because there aren't any. You are at your resort or bungalow and that is it.
This might be one of the few places in the world where I think a backpack is actually going to make your life easier here. There are a lot of places that have no footpaths or pavement so if you rock up with your rolling suitcase you better be prepared to carry it!