First up, I love to run, so I've never really felt like that. I think the big question is whether you are running because you want to run, or whether you're running because you feel like you should to lose weight. For me, I'd always wanted to be able to run, which is what gave me the motivation to keep going when it was hard to start with. But if I was just running because it was exercise then I might have considered switching to a different sort of exercise. Even if it doesn't burn quite as many calories, if you're likely to do it more often then that has to be a good thing. It may be that if you get into the habit of exercising you come back to running and really make a go of it.
I'm not telling you to stop running, by the way, just really trying to get to the bottom of your motivation.
If you DO want to run, but just feel self conscious when you're doing it, can you change when and where you run? If you're running on a treadmill in the gym, is it possible to find a quieter time until you get used to it. It's unlikely that anyone is looking at you thinking anything other than "good for her, getting herself fit" (if indeed they're looking at you at all, they'll be too busy with their own routines). Alternatively, just ignore them, because they're probably ignoring you
If you're running outside then you'll probably find other runners more encouraging, they'll say hello or smile as you go past. And as for people who aren't running, well, what do they know? At least you're getting out there to try!
If it feels physically uncomfortable, just remember to build up slowly and gradually. Don't try to do too much at once until you're used to it. You don't need to run for 30 minutes straight away, 1 or 2 minutes is fine to start building up. Don't worry about people seeing you taking walk breaks, we all have to start somewhere.
When I started running on the treadmill I used to see all these people with their treadmills set far faster than mine and wondered what they must think of me plodding along. Well, now I reckon I could beat most of them if I wanted to, it just takes time and perseverance.