It is a next action with no project. They are "allowed" in GTD.ToDo = Clean House. This is a recurring chore that I don't see as part of a project. It is not really a one-off item, because as soon as I complete it, it needs to incubate for about 2 weeks and then it shows up again. Is it an Action with no project, or is it something else?
There are lots of recurring maintenance tasks that are like this: fertilizing the yard, getting a hair cut, getting your car inspected. I use my tickler file to control these kinds of things. To take your example, assuming I would be cleaning every other Saturday, I would put a tickler card in the file for this Saturday. On Saturday, when I emptied the file, I would put the action "Clean the house" in my House action list, and then refile the tickler card into the folder for the Saturday after next. I would then clean the house some time that weekend. Two weeks later, I would get another reminder.
I would handle your exercise example differently. First of all, I would indeed classify it as an errand, because the only thing you do at the YMCA is exercise and you make a trip to go there. And Andrew has rightly pointed out how you could become more efficient by combining your trips to the "Y" with other errands. Considering how often you exercise, you might just want to control it with a daily checklist. One of the checklist items might be "Exercise?" If you need to exercise, put it on your "Errands" next action list. If you exercised yesterday or you are still carrying the next action from yesterday (or the day before, or the day before that), then you're already set.