Who cares if people make fun of you? If they do they aren’t people compassionate enough to be a good friend. I wouldn’t value the opinions of others who make fun of others for crying, having an anxiety attack, or struggling with something.
I feel so strange telling you what you “should” do so please do not take my advice as law. You are already doing something I would suggest and that is talking about this. You are reaching out and talking. Do you realize how brave that is? I believe that talking/opening up is the first step to recovery so you are on the right path :) So please, keep talking. Explain. If not to me (you can talk to me whenever!) than there are chatrooms online were you can talk and ask for advice in a relatively safe environment.
If working out is something you truly want then working out while at home is the only option I can see. There are workout videos on youtube. You can either participate while watching those or copy those moves while listening to happy/uplifting music. Even if it is only for five minutes today. Five minutes tomorrow. That’s okay. Condition yourself by making working out a fun, positive thing! It really can be a lot of fun. And maybe with talking about this and doing what you want to do you can realize that you should be so proud of yourself.
Be able to forgive yourself. It is okay if you can’t work out in a gym. You are a valuable, irreplaceable individual; and having anxiety or fear does not change that. You are not alone. You are loved.
Talk about your triggers. Know your triggers. Forgive yourself. Understand your comfort zone and don’t apologize for it.
If you need someone to talk to I am here, but these sites are helpful as well: