@Barrister This is long but probably something useful to you
I've been into the gym and around it's culture for over a decade now. I've approached women at the gym a total of 4 times, plus some spontaneous interactions that I wouldn't count as approaches. I'll give you some advice by sharing the stories of the 4 I did approach.
1.First ever real approach was almost 4 years ago, there was this hispanic woman that I started seeing regularly and she had the most amazing ass I've ever seen in my life. I really wanted to approach her but hesitated because it wasn't something I had done before but I ended up doing so when by chance we were both in the warmup area by ourselves. I walked over to her and just asked her what her name was and we spoke for like 10 minutes and I let her go back to her workout. I saw her a few days later and she positioned herself near me and when I approached she was really open and said how she had no plans later and asked what I was up to. I asked for her number and she then said that she had to tell me she was seeing someone, which I wasn't expecting. I just told her if she found herself single I'd be interested in going out and left. After that she'd approach me at the gym, touching me, etc. This went on for like 3 months and she ended up asking me out but reminded me of the boyfriend. I didn't follow through with it and after that we stopped talking and by chance she no longer went at the same time.
2. The other one I approached was around the same time as the story above in a similar manner, just walking up and asking her name. She was another hispanic girl from CA who was really short but had a great ass as well. This one was a little more awkward and not as open but would go out of her way to engage with me. However, something about her felt off to me on an intuitive level and I decided not to pursue.
3. This was from summer of 2018. There was another super hot latina that started going to the gym at my time and I decided to approach her, she was in between one of her sets on the leg raise station. I just walked over to her and waved, she took her headphones off and I asked her what her name was and we spoke for like 10 minutes before letting her get back to her workout. I saw her a few days later, she waves and comes over to talk to me and we discussed things we liked to do and hiking came up. She said she was free later and wanted to go, we lived in a similar part of town, and we exchanged numbers. She ended up ghosting me for a couple weeks and long story short it just ended up with her leading me on and being an attention *****. Months later I found out she did this to literally dozens of men regularly and bounced from boyfriend to boyfriend like a ping pong ball.
4. Fall of 2018. There was this light skinned black girl I had seen there for a while who always stared at me and would make extended eye contact, so I decided to approach when we were in the yoga/aerobics room by ourselves. She was a bit awkward but I asked for her number and told me she had a boyfriend, to which I replied saying it was nice to meet her and walked away. She literally would go out of her way to say bye or hi to me, approach me, etc. This went on for a few months and I asked her out again and she reminded me of this supposed bf, she never had one, and she literally didn't understand why this was happening. At the time I'm thinking why this woman has been going out of her way to talk to me just for attention when I've clearly already asked her out and expressed interest. After that she stopped approaching me and we just went about our lives, though I could tell with her body language she felt awkward whenever I was there despite it being entirely her fault lol.
To sum it up, it's best to try to get them when the gym isn't super busy and when they are doing something like stretching, warming up, in between sets, etc. I think it's also important to be able to pick up on when they aren't interested or if they are just leading you on/attention whoring, as it can rather easily turn into a very awkward situation if you go at the same time.