It's only weird the first couple of times you do it. After that, it will be your thing. People get used to things by seeing it frequently. It's like if someone wore a really stupid hat to work. The first time you see it, you think "Hmm... that guy is strange". After a while you still think the hat is strange, but it's normal in the context of the person. There may be people who discriminate against you, but likely the damage will be small (especially for something like push ups).

In my last work me and a coworker started the 100 days push-up challenge. Soon our whole team joined. After a few weeks many people from other teams also joined and it became a thing in the company.

At my workplace we have a room with gym mats, wall/pull-up bars, a kettlebell and other things like foam rollers. Should be the standard everywhere.

Just step outside and do them on a sidewalk, it only takes a minute or two.

I keep a pair of tight-fitting lightweight mechanix gloves on hand for doing pushups anywhere...

Squats in the bathroom!

Hearing that makes me sad.