Dear Wise and Knowledgeable Jono,
I'll just keep this short...
Is it possibly to start a relationship, like a romantic one, with a guy without flirting?
Because I am a girl who fails miserably in that category, possibly because of my fear of rejection.
The answer here kind of depends on your definition of "flirting." You're going to have to do something that carries some risk of rejection, even if it's just ordinary conversation. I suppose you could sneak into a room with a hot boy in it and just move so incrementally slowly that he didn't realize you were in motion, until eventually you had your arm slung over his shoulder and were making out with the side of his bewildered face, but I rate the likelihood of this working at 0.9 Jonos (where 1.0 Jonos equals "catastrophic romance failure").
If all you want to do is skip the actual approach part, but you're okay with him approaching you, sure, that's possible. If you would like to skip the whole awkward beginning stages, including the "I like your, uhh, pants, let us go out for a dating meal of uncomfortable spaghetti," that's also possible, but it requires a friendship (or at least a good acquaintanceship) first.
1.) "The Look"
If all you want to do is get approached without doing the approaching, then Auntie Sparknotes has already written a guide for you. I am directing you to hers instead of advising you myself because my take on this would be "Look at him with your eyeballs, no, the other part of your eyes, flip your hair turnways, bluh I give up." Everyone would go be so busy winking and hair-twirling and grimacing in confusion that I would create a generation of unmarriable teenagers. So just do what Auntie says.
Whether or not you count this as flirting, it is the best way to get the average guy to approach you. If fear of rejection is your only problem, this should help, because once he's approaching you, you've already been selected, and the only way to turn that into rejection is if he approaches from behind and taps you on the shoulder and you shriek and throw a tray of pies into the air and they all land on his hapless face. Now that I've jinxed this scenario for you, it will definitely happen. Sorry!
2.) Have a totally platonic conversation
If you've gotten the dude to approach you, this is a safe next step; it's also good practice if you're just trying to get to know a guy better. I always found that first dates were much less horrible if I had a semi-friendship first. In my experience, it gives you common ground to talk about, when you would otherwise just go "So you're a girl, wowzers (voice cracks) aw jeez" and hide in the bathroom.
Either way, for those of us who suck at flirting, the point of this early conversation is not romance. I know I've often said that if you so much as sneeze on a high school boy he will take it as a sneeze of proposed marriage. But there is such a thing as regular old conversation, not composed of comments about how he has a cool shirt or a sextacular face or whatever. Don't try to compliment him or convey interest; don't even try to flirt at first. Just have the same kind of neutral conversation you'd have with anybody else. Learn stuff about him. Be all "Psh, (x) sucks," where X is a thing that sucks. You might do nothing but exchange names and say you'll see him around; this is okay, because it sets up later interactions, and in the future he won't react like to you like you're a crazy lady leaping out of a nearby manhole.
(If you're already friendly with a guy and are trying to woo him, these conversations will have to eventually develop in a more flirty direction, but it really doesn't have to be too painful.)
3.) Be aware of your dumb face
To clarify, I have no problem with your specific face, except for the fact that the muscles controlling it are unaware that you're trying to attract a boy. I never realized how poorly my face communicated emotions in high school until I tried to joke around with one guy and he thought I was literally going to fight him. Some of us frown a lot, some of us glower; I scrunch up my eyebrows like I'm trying to do calculus, even when my dating life involves very little calculus. My point is that these expressions communicate totally different things, even if the exact same words accompany them:
Look at yourself in the mirror as you say "Hi, what's your name?" If you look curious and interested, then you're fine; the last thing I want to do here is make people self-conscious about their faces. It might seem shallow to worry about your appearance—and if would be, if I were telling you that you require a specific butt to be attractive to gentlemen—but if your default expression is "mad about calculus," this is going to get in the way of your non-flirting flirting. Whether you want a boy to swoon over you and ask you out, or to become friendly enough that you can comfortably take it to the next step, the common element is appearing welcoming and approachable, instead of appearing to have just eaten a bug.
Are you afraid of flirting?