It's incredibly difficult to prevent a wooden ship from leaking completely, but leak rates can be made quite manageable. Historically, any wooden ship making a transatlantic voyage would have had several people on duty (typically through rotating watches) at any given time, and some of the time some of the crew would be performing maintenance related to the leaking of the boat. This might include inspecting the bilge (the lowest part of the boat) where any water would naturally collect. And, of course, bailing or pumping out the bilge on a regular basis. Both of these are activities that occur typically at least daily on basically any boat. Bailing requires no technology beyond a simple bucket, bowl, cup, or any other type of container of liquid. Bilge pumps have been around since the middle ages, and were in use on some ships during the "age of discovery".

The other side of the equation was constantly inspecting the ship, identifying leaks or gaps, and plugging them. One common material for that task was "oakum". Oakum is a material made from hemp fibers, typically picked out of old ropes by sailors when they didn't have anything else immediately at hand to do. It would be bunched up and pounded into gaps or holes using implements similar to blunt chisels or nails (for linear or round holes) then it would be sealed up by applying melted tar or pitch.

This combination of activities, which would be ongoing and occur basically every day throughout the entire operational lifespan of the boat, would keep the rate of leakage small and the buildup of water manageable.