Yes, Major League Baseball does have drafts, they have two drafts, they have the First-Year Player draft that's held in June which is when all 30 teams draft players that are either in High School or college or have already graduated.
The other draft is the Rule 5 draft which is held in December and teams that have space on their 40-man roster are allowed to draft eligible players from other teams that aren't on their 40-man roster, teams are not required to select a player however in the event of passing then they will not be allowed to select any players in that draft.
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You can be drafted out of high school if you are very talented. It is rare and you usually start in the AA or AAA first then move up even if you are already drafted officially to a team.