This has to be done on Arcade Mode on any difficulty. Your task is to make sure that you do not receive a single hit while working on defeating your opponent.
To earn this achievement at a quicker rate, put the difficulty setting to Very Easy and set the round limit to one round only. Then chose your favorite character, take on the first guy you see, and be very aggressive. If they come back up with a quick hit, make sure to stay away from it or block it if you are fast enough to anticipate their move. Then continue working on them until you've beaten them inside the ring or taken them out of the ring. If you manage to do all this successfully, you'll win the match without getting hit and you will earn your achievement.
Have fun and good luck! If you see fit to submit a negative vote, please tell me why and I'll happily correct my solution to make amends on my behalf. Thank you!
Bryan2041 offered the following tip for this solution:
You only have to win a round without taking damage to get this, not the whole match. The easiest was to do this is set the match to very easy, then pick the first character ( the sumo guy I don't feel like looking up his name ), then tap B. This should also eventually get you the Ring Out victory.
Credit goes to Bryan2041 for the tip! Thank you!