Start potty training him!!! Get yourself a potty chair and put it on the floor in the bathroom near the toilet. Everytime you go or Dad goes, he goes. You sit and read a book together. You buy pull-ups or thick cotton training pants along with sweat pants or another type of elastic waist banded pants that make it easy for him (and you) to push down and pull up when he needs to go to the bathroom. You reward him for a job well done and consecutive days without a day time accident. Hurrying to the bathroom and not quite making it is not an accident, he was trying to get there!! Two year old kids are more than ready to be potty trained. You just have to provide them the tools for success. And yes, it's going to take extra work on your part, like more laundry for a while, but it will be a limited time frame. Cotton training pants work best because it's uncomfortable when they're wet, and nothing trains quicker than discomfort. Little girls do really well with rayon/nylon panties, they're pretty and they are extremely uncomfortable when damp.
Model the behavior you want them to have, in short, share the bathroom with them. Little boys at this age don't need to stand and aim in the toilet, they aren't tall enough yet to clear the toilet bowl. A potty chair on the floor where they remain seated for the entire performance is best for you and them, success is achievable.
When your child has outgrown the diaper sizes available, it's definitely time to potty train.