It sounds like you think the problem is that he is generally distractable. That might be true, or also he simply might have difficulty connecting one step to the next when completing a goal (such as, put on your shoes so we can go to the store. Putting on shoes can still take a while even for a kid motivated to go somewhere.)
But it also could be simpler than that -- if he's mostly doing this when directed by you to do something around the house, he might just be resisting the interruption of the flow of his thoughts and think the things you ask him to do are not interesting. Presumably if he refuses point-blank to do a task, that would bring down a negative response from you, which he doesn't want. But some kids learn that they can still support their own autonomy even when a parent is telling them to do something, by dragging their feet, doing other things along the way, etc. His (nominal) movement in the direction you have told him to go will keep you from being mad, but he also gets to be in charge of himself.
My suggestion would be to see if he gets this distracted on the way to something fun he likes. If he gets this distracted when doing something he wants to do, not just when doing something kind of boring that you want him to do, it seems like it might be a tendency to keep an eye on.
If he can't seem to keep his focus on anything at all, fun or not, you might take the time to break the tasks out, give instructions one small step at a time, and like that. If the problem of distraction seems extreme, sometimes professional help (such as occupational therapy) even comes into play.
Frankly, though, the example you gave sounds more like garden-variety kid resistance when told to stop one's own activity and do a task at Mom's request. Yes, he needs to know that as part of the family he has to help out, but no, he shouldn't have to do it at the drop of a hat when he's in the middle of something else. Give him a time frame, or set a timer, and be considerate of his autonomy. He probably wants to do the things the members of the family depend on, just be careful not to expect him to always leap up whenever you ask him to, like he might have been delighted to do when he was 2. :)
The short form of what I'm trying to suggest is to be sure to solve the right problem. If he's purposely dragging his feet to feel like the boss of himself when the task is dull, solve for that. If the problem is that he cannot maintain his focus on anything whether it's fun or not, solve for that. I'd say it sort of sounds like the issue is more the first one.
Yes, I definitely agree with specialmom...particularly the motivation part. Kids of this age really haven't learned much of a "work ethic" and would rather do something (anything) fun. Also the smarter a child is, the better they problem solve ways to avoid doing what they don't want to do. Unless, his teachers are also concerned, this is a "habit" only if you allow it to be. Try giving time limits to get things done. Maybe set the timer on a smart phone...just be realistic.
That's hard. I have one child who is especially like this. Six is a young age to worry that distraction will be a life long issue. I think at that age, the motivation to something like "go get your shoes for school" is low when they may really want to stay home, you know? But as they get older they get more motivated for things that they want to do or see consequences (like getting a tardy from school) to be something that spurs faster action. I would take a positive approach. Praise like crazy when he DOES do something and stays on task. Do his teachers report anything at school with this?