My Apple Watch Ultra 2 Side button activates emergency services menu while working out.

I use the water drop feature (aka "water lock") - I turn it on before working out and off when I'm done. This has helped me lock the buttons from inadvertent presses. More here: How to use Water Lock and eject water from your Apple Watch - Apple Support

This happens to some users when they bend their wrist and press against the Apple Watch. Either the wrist or the clothing they are wearing.

First try changing the "Side Button Click Speed"

Settings -> Accessibility -> Side Button Click Speed “Side Button Click Speed” can adjust the press and release timing, so you get the Control Center display every time. Adjusting The "Side Button Click Speed" is also useful for Apple Pay’s double press and release. There is “Default” (sub second timing of needed), “Slow” (up to a second between press and release), and “Slowest” (up to 2 seconds between press and release). I like the “Slow” setting. NOTE: Slow and Slowest settings do not mean you have to wait a full second or 2 seconds between press and release. It just means you do not need to be super fast.

The other thing you can do is change the watch bands so that the Digital Crown and flat oval side button point towards your Elbow. Then use Settings -> General -> Orientation to switch the watch face around so it is properly oriented for reading in the new orientation.

This is a great tip Bob. However, as of Watch OS 10 Apple has combined this setting to include both the side-button and Digital Crown. I have also tried switching the orientation and find that works best, but it also feels a little awkward with scrolling.