Question:Was Naomi Obed's grandmother or even related to him in any way?
Obed was the son born to Boaz and Ruth. Ruth was Naomi's widowed daughter-in-law. She had married Naomi's son who had died. Boaz married Ruth in part because he was a near kinsman to Ruth's husband. In particular he was the second nearest kinsman (Ruth 3:12).
Many assume that Boaz was one of Elimelech's nephews, meaning that Salmon was Elimelech's brother (Ruth 4:18-22). However, there is no way to prove this. So if you are looking for blood relations, they are not close.
However, in society one can function as a relative. Ruth's oath basically made her more than a daughter-in-law to Naomi (Ruth 1:16-17). Naomi refers to Ruth as her daughter (Ruth 2:22; 3:1, 18). Even the neighbors realized their close relationship. "And may he be to you a restorer of life and a nourisher of your old age; for your daughter-in-law, who loves you, who is better to you than seven sons, has borne him" (Ruth 4:15). And Naomi helped to raise Obed (Ruth 4:17). So while not a blood-relation, Naomi still functioned in every way as Obed's grandmother.
We do the same today. If a woman becomes a stepmother or a man a stepfather, we still refer to them as a mother or father and later in life as a grandmother or grandfather. When children are adopted, they still refer to the ones who raised them as their father and mother. So what is spoken of in Ruth is not unusual.
Most importantly, however, is to go back to the reason why Ruth and Boaz married. Ruth was the widow of a man who died before they had any children. Since their marriage was to raise up an heir for Ruth's dead husband, that first child would be considered the child of her husband and, thus, his heir. "If brothers dwell together, and one of them dies and has no son, the widow of the dead man shall not be married to a stranger outside the family; her husband's brother shall go in to her, take her as his wife, and perform the duty of a husband's brother to her. And it shall be that the firstborn son which she bears will succeed to the name of his dead brother, that his name may not be blotted out of Israel" (Deuteronomy 25:5-6). Thus, from an Israelite's point of view the child was the dead man's and thus would be Naomi's grandchild.