Would you just try a new sport and meet people there? How would you even find people that do a certain sport?
You can definitely find friends at the gym, find someone of a similar size to you and ask them to be gym buddies. Don't weird them out, make sure to shower and get your hair done before =profit?
Invite them out, accept their invitations. You don't have to become friends. Get to know their friends and friends' friends. You'll click with someone, start inviting them out to things or over and accept their invites.
Create opportunities for people to be social with you. Go to a bar or restaurant that hosts trivia, attend regularly. Chat with the other teams, invite people you meet to join you.
If that doesn't interest you, find other activities. Go to movies (with dinner before/after since movies aren't really social), host a movie night, board games, set up a hiking group, or find one to join.
One thing I've learned, I have to take the initiative to invite people. And I can't turn down invitations when I've first met people. If I meet someone and sort of hit it off, but turn down their invites to activities, I'm not going to be invited to join them again (or I'm less likely to be). But when I invite people, I'm generous about re-inviting people who turn me down. I throw out invites to 10-30 people for events, and I may only get 5 to go.
At some point the relationship will become a real friendship, not based on doing activities together. But you have to start somewhere.
But on a more positive note, from everything I've done I also think this kind of positive, magnet-like attitude is the way to go. It's almost like you have to go cross ways with 500 people in order to squeeze out 1 or 2 friends, so you need to speed up that process. Is that how you feel?
And yeah, sometimes it can be demoralizing when people don't show to something I've put a lot of time into. But that's also why I've learned how to find easier social events to plan. The town I live in does a First Friday thing (more live music, art exhibits, etc.). I just throw the invite out to a bunch of people, if they show they show. Same for movies and other things. At some point a small "inner circle" or whatever forms, that's the group that'll stick around for a while. Some may just be more into going out, others will be close friends. But at least you've got the group started. It'll grow from there, especially if you can find a couple more socially outgoing people who enjoy planning.