If you want to keep the carrots fresh then refrigerating the carrots is a must.
Carrots are crispy root vegetables and if they are not kept cold they will get soft and limp after a week or so and sometime sooner depending on the climate.
If you're gonna eat the carrots within a few days of buying them or picking them from your garden then there's no need to refrigerate them.
I usually just pick carrots from y garden and use them up as I pick them but sometimes I'll pick the rest of the carrots and then refrigerate them or store them in a dark cool place in my basement.
People used to long time ago store vegetables in dark cellars which helped to keep them from going bad and makes them last longer.