Set in the medieval period, preferably around 15th century, there is a habitable island the size of New York surrounded by boiling water inside the mouth of a volcano. How could you explain that there is a small human settlement on that island? Whether the volcano will one day blow itself or not is a topic for another day ;p
I can imagine such a refuge being very nice in winter. The discoverers were trapped out in the cold with no firewood and so made for the volcano in hopes of not freezing. They could rappel down from a height on a rope tied to a big rock which they threw down onto the island. They dug little channels in from the boiling water and made hot tubs where they soaked the winter away like Japanese snow monkeys. Scurvygrass, dodo birds and the aforementioned excellent weed is all you need for paradise.
Your island sounds a lot like this prison that is used in the TV show Avatar: The Last Airbender. In the show, they accessed the prison using a pulley and gondola system, but the methods mentioned in other answers could have worked as well.
Perhaps in your world, there was a former civilization who used the island as a prison, but abandoned it at some point. The settlement could be made up of the former prisoners who either couldn't or didn't want to leave.
If they do not leave or enter the island all the time, it could be that an earthquake activated the volcano. Each earthquake may change the situation.
The boiling is not too much of a problem for a big wooden ship. Inside the ship it remains fairly comfortable, for a while. A storm or an extremely urgent need to get on shore could be a motivation to do it.
The most straightforward solution would be essentially a single-use drawbridge. On the outer edge of the water, build the bridge and two towers. Anchor the bridge at one end, and use pulleys in the towers to first pull the bridge upright, then lower it across the water. Once the bridge is in place, the other end can be secured and (optionally) the towers and pulleys can be disassembled. Counterweights could be used for a particularly large bridge. If the water is too wide, a series of bridges between conveniently-placed smaller islands could be used.
Since this is the same principles behind a castle drawbridge, which was a common defensive structure in the medieval period, the technology would be no problem. In fact the technology for lifting much larger and heavier objects vertically and lowering them back down was pretty common long before medieval technology was available, at the very least the Ancient Egyptians, the makers of Stonehenge, and the Easter Islanders could all do it.
There are other similar solutions. They could rotate the bridge horizontally while supporting it with ropes. They could also just place two long tree trunks, either by lowering them with a simpler pulley system, rotating them, or if they are much longer than the water is wide just pushing them across. The planks could then be installed one after another, having the workers stand on the previous plank as they secure the next one. These sorts of solutions might be a bit simpler if the bridge doesn't need to be raised and lowered.
A better question is why they would want to live there. You could argue that this is a mining town, harvesting some rich mineral or gemstone resources released by previous eruptions. You could say someone spotted the glint from exposed gold or gemstones, and that is why the bridge was constructed.
Once every X amount of days/months/years, the volcano calms down, the boiling stops, and the island becomes accessible for a short time. A bit like the Old Faithful geyser, but backwards. Instead of erupting periodically, it stops erupting periodically.
You could require that both the volcano timing and the tides need to be correct, making it harder to predict and with possibly much longer and seemingly irregular intervals. Perhaps the weather also plays a role. The original explorers came across the island by chance in a calm period, and wandered in, and got trapped there. Or that the steam provides excellent defense, and they are happy to live in there for X months/days at a time since they'd be raped and pillaged by bandits otherwise. No one else has figured out the timing pattern, so it seems like its always inaccessible. Perhaps there's excellent weed growing naturally there. I think the problem is not to explain the steam and boiling (its the mouth of a volcano after all) but why the inhabitants are actually there in the first place.