There are several halachic aspects (pertaining to Orthodox Jewish religious law) which apply to “walking” for exercise on shabbas. When my neighbor heard that I “walk” on shabbas, she noted that the practice was forbidden. Actually, it is not so simple, and as I explained to her, I do not *REALLY* walk on shabbas- my walking on the Sabbath day is slower (not strenuous) and generally of a lesser amount. And I don’t go out “walking” if I don’t feel like it.
Rabbi Neuwirth in "Shmirath Shabbath Kehilchatha" writes that he heard from Rabbi Shlomo Zalman Auerbach, shlita, that if the person enjoys the exercise it is permitted for him to do it on Shabbat. And that is the main reason I walk on shabbas--- walking makes me feel better physically AND mentally. The fact that I know that it is good for my health is irrelevant.
However, there can be other problems. If one walks outside of the area of the eruv, one can not carry drink, tissues, or the like. Also in non-religious neighborhoods (but a Jewish community), one needs to be careful when crossing a street not to cause a driver to brake because of you. And sometimes this is much easier said than done. Often the only way is to stand aside and appear to not want to cross the street at all, and then to dash across when no cars are coming--- which yesterday, at one corner, caused me a several-minutes-long break.
Another problem with exercise "walking" on shabbas, is that if something unforeseen happens, you may be stuck a good walk from home, and no way to get back there besides walking back. For example, yesterday for the first time in ages, I went walking in a non-religious area. I had a sudden urge to walk over to the national park, and since I REALLY felt up to it, and wanted to go, I did, although I took care to walk slower than usual. When I was a good 45 minutes away from the house I slipped and fell (the pavement was wet). Luckily I was not hurt, but as I got up I was initially horrified to realize that I had slipped on dog droppings. I panicked, thinking that if my clothes had gotten dirty, I was stuck to remain that way until I returned home. LUCKILY my clothes were clean, and I was able to continue on to my destination. Subsequently, on the way home, a second “unexpected” occurrence popped up. The beautifully sunny sky became clouded, and a downpour started. I dashed under the nearest building for cover, and after 5 minutes the rain stopped, and I was able to continue home. So if you are a Sabbath observer, and you “walk” on shabbas, BE CAREFUL!