Yes, as Fox said you need a tavern and a cook, but also note you need peasants to carry food from the storehouse to the tavern as well, or otherwise the cook will just sit there and do no cooking and your people will starve .

Actually peasants are the most important job in the game since without them to cart things from one place to another nothing gets done, I usually reckon on having half of my population as peasants most of the time.

@Conor, I personally tend to only set everyone as builders when I actually need something building and then go back to having peasants, since that way I get something built as quick as possible.

Indeed, in general the best strategy for  Castaways I've found is to have one or two perminant jobs such as cook, but then chop and change as needed, e.g there is no point having a doctor unless people are injured since the doctor's secondary job of tailor isn't that useful, also to make sure to get to know what people's secondary jobs are since often you can do without a full time job if you've got people doing jobs part time, for example I don't tend to bother with a full time metallurgist once I've got a smith since the smith can be busy smelting when they're not smithing.

You don't just need to build a tavern, you also need to assign one of your people to be a cook, to prepare meals and more of your people to be peasants to take food to the tavern.

That should stop your people from starving for long enough for you to start getting food production up and running.