Is kendo played inside or outside?

It doesen't really matter wether it's practised inside or outside (generally it's practised inside), as long as you get the proper floor. The place where you practise Kendo is called a dojo (道場); doujou roughly means "place of the way" since modern Japanese Martial Arts are usually called "do" (道; way), the place wher it's practised is the "place of the way".