Baseball was not, despite certain legends, invented de novo by Abner Doubleday (who never advanced any claim to being involved in its creation). Its rules instead come from a game played in the state of New York in the middle of the 19th century, which in turn derives from the same underdocumented medieval precursor as cricket. (There are in fact high medieval manuscript marginalia showing people swinging what appear to be modern baseball bats at round white balls.)
But the New York game was just what caught on beyond the region of its origin, because of the clout of New York City during the formalization of America's national sport. Other local variants existed at the time, the most prominent being one played in and around Boston. The Massachusetts version of base ball at that time (now usually known as the "Massachusetts game" but commonly called "town ball", "round ball", or "base" by its players), formalized by the Massachusetts Association of Base Ball Players under the name "town ball" in 1858, was different in several ways.
In retrospect, many observers consider Massachusetts town ball to have been a superior spectator sport in comparison to its New York counterpart. Some of the features that would have made it so have since been adopted into the national version of the game: for instance, catching a fly ball for an out worked somewhat differently, such that in the New York (and early professional) version of the game it could be done on the bounce as well. The modern rule was preferred in Massachusetts. Similarly, round ball required overhand pitches, whereas the first professional rules of the national sport forbade them (as softball does today).
On the other hand, proponents considered the combination of a wide open field and closer quarters a key strength of round ball that was lost as it came to be displaced by the national rules. When the rules of town ball were written down in 1858, the bases were set only 60 feet apart, rather than the now-familiar 90, and the striker (batter) began halfway between home base and first base (so the pitcher was throwing toward a side of the square, rather than along a diagonal) and so only had ten yards to go before being safe on first. Meanwhile, there was no edge of the field, no foul territory (any hit was fair no matter where it went), and no requirement for the runners to stay within a base path or infield. It's likely, had this caught on as our national sport, that spectators would have been seated rather further from the diamond than in the baseball we know today.
Hits were also more common thanks to a differently-defined strike zone: the ball was to be thrown within reach of the bat, on the striker's preferred side. Any ball the striker failed to swing at was ignored, though if they were thrown within the correct region and the player didn't swing, this was considered a way of delaying the game; doing this repeatedly would result in a strike being called as a penalty, but this penalty did not apply simply to the first time a player didn't swing. Furthermore, for a strike to count, the catcher (or any other player on the fielding team) had to catch the ball; three uncaught strikes (wild pitches, in effect) were counted as a hit, allowing the striker to begin running the base path.
To deal with catching runners in this more challenging open field with more hits associated with it, two other rules were in place: first, teams were larger (with ten to fourteen people in a batting order and fielding side, rather than our familiar New York style nine), and second, runners could be "soaked" - struck by a thrown ball while not on base for an out. Teams also exchanged places after each out, rather than every three, which likely meant the bases were more sparsely populated. Despite these rules, the game heavily favored hitters. Rather than ending after a fixed number of innings, town ball was played until one team had racked up a hundred tallies (runs), and these games were nonetheless not considered remarkably long.
The full 1858 rules of town ball are preserved online by Baseball Almanac, for those wishing to relive our national sport as it could have been.