Do they need exercise? No.
Can they become obese? Yes, but with that you just need to keep an eye on how much and how often you're feeding.
However, they do need adequate cage size, regardless of the fact that they don't need exercise. For terrestrials it should be about three times the legspan of the tarantula in length, two times the legspan in width and one to one and a half times the legspan in height from the substrate to the top of the enclosure. For arboreals it's different, and as long as they have about two legspans in height for the enclosure they're fine.
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