How did people from Ancient Greece keep their bodies amazingly fit?

Keep in mind that first of all, most ancient artwork of the human body tends to accentuate the ideal form--the young, healthy, well-fed, and athletic. It would make sense that not many Greeks would want to decorate their homes, temples, and public areas with something which depicted malnourished, unattractive individuals. We have plenty of evidence of how highly the Ancient Greeks valued physically fit and healthy male and female bodies as ideals of beauty, and much like today most decorations depicting the human body would no doubt display this ideal. So from the get-go, the fact that we have tons of statues, reliefs, paintings, amphora, frescoes or what have you from Ancient Greece of exemplary human physical fitness, this does not mean that all Ancient Greeks themselves were exemplars of physical perfection and beauty. The harshness of pre-modern life means that there were certainly many people suffering from afflictions, diseases, and malnutrition that we don't see in those Adonis-like statues.

That being said, we do know that pretty much all of the Ancient Greek cultures placed a very high value on their standards of physical fitness and health. Athleticism and fitness were highly stressed and highly valued, not only for reasons of health and attraction, but even as a reflection of good morals. Young men and women were highly encouraged to be active and athletic, taught from a young age that this was not only an ideal of beauty but also a part of their civic obligation to be valuable and capable citizens. Women had to be healthy in order to raise healthy children--men had to be fit and strong in order to be able to better serve and protect their families and communities. Both men and women practiced many sports and exercises from a young age--running, wrestling, discus and javelin throwing, rowing, riding, etc. Weight training has also been well recorded among Greek athletes, who competed in popular Ancient Greek athletic competitions which stressed individual physical fitness and accomplishment. Many of the Ancient Greeks were also well known for stressing the values of temperance. Sparta is of course the most famous example wherein male citizens were subject to an incredibly strict martial lifestyle from a very young age, and female citizens were likewise made to engage in regular and vigorous exercise--not to mention the practice of abandoning sickly or deformed babies at birth to the elements in a sort of ancient policy of eugenics.

Greece itself, given its climate, environment, and extensive trade networks allowed for a relatively diverse and nutritious diet. Fish, grains, fruits, olives, etc form the basis of a varied, healthy, and nutritious diet for those who could afford such variety. These wealthier citizens were no doubt also the ones who are typified within Ancient Greek records, and thus we see constant examples of the healthy Greek citizen. Little attention meanwhile, is given to portraying the appearance and health of say, an impoverished Helot living under the cruel yoke of Spartan tyranny--one could easily imagine them suffering from far more malnutrition and physical ailments to say the least.

And lastly, it depends on what you're comparing it to. If you're looking at the issues of poor physical fitness today vs that of the Ancient world, then you're looking at it the wrong way. Today our main issue regarding poor fitness and nutrition is that many people are too inactive and overweight--that kind of poor health was a luxury only the very wealthiest people in the entirety of the Ancient world could afford, both in Greece and beyond. Not only was it far more expensive and difficult to obtain such excessive amounts of calorie-rich food at the time, but also bear in mind that prior to the industrial revolution life involved way more physical activity, period. Going somewhere involved walking or rowing for the vast majority of people, building something meant doing everything by hand or at best with the help of some sort of animal for certain things. Getting water meant hauling it out of a well. Life was violent, and violence meant physically overpowering your opponent and being capable of doing so--not just pointing a gun. Most people were farmers, some were sailors/fishermen, and a small few filled the rest of the occupations--modern farming is still hard labor today and it was multitudes more difficult back then; sailing and fishing involved backbreaking work hauling oars, ropes, and nets all day, no engines involved; crafting tools and such meant slaving away with your own hands and your strength to fashion said implement as well as hauling all the supplies necessary. I mean the examples are endless. Physical fitness in terms of low body-fat was simply the norm for all but the wealthy few who didn't have to labor. However, what we aren't as quick to recognize is that a lack of proper nutrition and caloric intake meant that many people, skinny as they may have been, weren't exactly hunky and toned fitness models. Nutritional deficiencies meant things like rickets, deformed bone growth, and stunted growth for many people. A lack of readily available protein meant that your average less-than-wealthy Greek may have been slim, but despite a life of heavy physical labor they weren't exactly adorned with heavily muscular physiques either--a poor laborer would appear to be pretty scrawny despite the lifestyle of intense physical exertion--you can't put on muscle after exercising if you don't consume enough protein. Such a life took its toll physically, and while it is a misconception that the average lifespan of ancient societies was somewhere in the 40s and 50s (high infant mortality rates brought it down, but if you survived to about 5 then you'd probably live to around 60-70 unless you were killed), the physical toll upon the body (bones, joints, skin, etc) meant that by the time you were 50 you were lucky if you weren't using a cane.

Long story short, Ancient Greek Citizens (the citizens being the wealthier, landed members of society aka middle/upper class) had access to relatively diverse and nutritious diets, lived lives which were inherently more active as it was in the entirety of the Ancient world, lived in a culture which placed a profound emphasis on physical fitness, beauty, athleticism, and temperance, and were generally the ones who were the primart subjects of Ancient Greek artistic portrayals of human beings and subsequently their physiques--artistic portrayals which in either case invariably sought to portray/capture a cultural ideal of physical fitness/attractiveness/health over genuine likeness of its real subjects. Meaning that there are reasons why some Greeks were healthy, but your question itself presumes a bit to begin with.