Yes, you can technically enlist in any state you wish. However, keep in mind that you will be expected to drill in the state you enlist in. Therefore, if you enlist in a state that is some distance from your home of record, be prepare to travel.
Although you did not ask the question out right, I'll add that the Army regulation on tattoos is as follows: The Army's tattoo policy (AR 670-1, Chapter 1-8E ) prohibits any tattoo or brand anywhere on the head or face, except for permanent makeup on females (such as permanent eyeliner, eyebrows and makeup applied to fill in lips). Tattoos that are not extremist, indecent, sexist or ****** are allowed on the hands and neck. During your initial entry exam, determinations will be made according to current regulations.