While there isn't a mechanic to tame rabbits like there is with dogs, cats, and horses, there is still a way to keep them as pets. Keep lots of carrots handy when exploring biomes to bring one back home.
Players can use either carrots or a lead to get them to follow. Using a lead is the preferred way, as carrots require more attention and slower movement. Rabbits can potentially become disinterested as well and switch back and forth between food to eat and the carrots to lead can be daunting.
Once back at home, players can create a pen, hole, or keep them in their house with a door. For pens, any fences or walls will do as long as no blocks around the perimeter can be used by the rabbits to escape. Keeping them inside with a door is simple as the player only needs to close the door to keep them from getting out. Holes in the ground can be used as well and only need to be two blocks deep to prevent them from jumping out.
Once placed at home either in a hole, fencing, or house, rabbits can be bred with one another. Using carrots or dandelions, rabbits will breed just like any other animal in the game.
To learn how to get even more pets, check out How to tame an Axolotl in Minecraft on Pro Game Guides.