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When I was a college student competing in track and cross-country running, I ran 7 days a week. I was lucky I didn't get injured, but your body can get away with more when you are 20 years old. When I started training for marathons after college, I researched some basic marathon training plans and all the ones I found recommended at least one rest day of no running. Some plans call for two rest days. Some runners like to take Sunday off after a long run on Saturday, others like to take a day off during the week when they need some extra time during the day for something else, or they make a spur-of-the moment decision to take a day off on a cold, rainy Tuesday.
Rest doesn't mean lying on the sofa all day, although that's fine. You can bike, swim, chase your kids, or go shopping. Just don't run. Those muscles that you use in running will know that they aren't being used to run, and they will recover more effectively, even if you are engaged in some other activity.