Lots of good info here -- NavyHoops especially is spot on with her honesty about the difficulty of walking on in "ball sports" in particular. Navy is a Division I program, and the teams are very strong overall. If you are wondering how you stack up (and this works for most colleges/universities where you might want to play a sport), take a look at the roster and/or press guide and focus on the honors and accomplishments typical players have garnered, whether it's all-State, or playing in an Under Armour game, etc. It will give you a sense of your competition. Because Navy isn't hamstrung by scholarship limits, teams like football tend to bring in a lot of recruited plebes so walking on as a "Rudy" like person is, as NavyHoops suggested, virtually impossible. Sprint football is a great option though (although still some very impressive talent there). Rowing, if you've got the right build (tall and skinny for the lights, tall and muscular for the heavies, or tiny and compact for coxing) is probably the best home for true walk-ons, and NavyHoops is right on there too -- close bunch of guys but many start, fewer stick with it.
At non-Service Academy Division I schools, you'll sometimes hear talk about "preferred walk-ons." Essentially, those are athletes who are not on scholarship, but the coaches are aware of them, want them to come out for the team, and there can even be a presumption that they will get a roster spot. If you think you've got the talent and drive to play at Navy, you can try to make yourself the equivalent of a preferred walk-on by getting in touch with the coaches ahead of time. Sending in times/distances, for sports where that's relevant, or maybe a link to your film on YouTube, if it's not. Check out their contact info first, don't be a pest, but maybe you can become somebody who is not just a face in the crowd but will get that second look. With that said, it's a long shot for a lot of sports.