I used to feel the same. Right from being a teenager, I never got invited to parties, sleepovers and nights out like everyone else did, although I never knew why. I was a little overweight and no Rembrandt, but then neither were the others.
It went on until just a few years ago. I joined a band and, whilst I have no real, bosom buddies that I socialise with, I have lots of casual friends. I have a number of similar friends at work. It seems that when you become absorbed in something and do it well, then people will want to be around you. Even so, the two crowds are so diverse I could never introduce one to the other.
As for a best friend - he's 8000 miles away. He's my cousin, two years my senior and I visited him three years ago after 26 years. As we talked over the beers on his back porch, he said he'd felt the same over friends - loads of casuals but no real soul-mate to speak of. In fact there were secrets that the two of us had only ever shared with each other, by letter. A joint realisation dawned that we were, in fact, each other's best friend. Just that being cousins, and of the opposite sex, the idea that we could also be best mates had never occurred to us.
They say you can choose your friends but not your family. But sometimes they can be one and the same.