There are two things to address in your question. The first is the idea that deodorant did not exist, the second is the idea that people naturally smell bad. As a note: my answer will be Europe specific as I know less about the personal grooming habits of other areas.

Deodorant is, in it's most basic form, a substance that uses one scent to cover another scent. It does not stop you from sweating or smelling, it just covers up the smell. What most of us call deodorant today is in fact antiperspirant, which blocks the release of sweat by blocking up the sweat glands and, usually, includes a fragrance as well. This product, indeed, did not exist for most of history but other, smell masking, products did exist. Things like scented talc powder were popular, as were small bags of sweet smelling herbs and flowers. Perfume is also a deodorant, in this sense, as it was often used to cover up less nice smells.

As to actual smell, it depends on the time and place. The first thing to note is that many civilizations in the past had bathing habits similar to ours. If we look at Rome, we see that bath houses were plentiful and very important. Cleanliness, as we consider it today, was indeed practiced by all except the extremely destitute. Some cultures, such as the Jewish culture, practiced ritualized bathing, while others had traditions of sweat baths or sweat huts, such as the Finnish and Russian saunas. What is important to remember is that cleaning of the body, whether with soap and water as today, with oil as the Romans did, or with a dry cloth and rubbing (to slough off dead skin cells and to absorb old oils) as was recommended at different times in history, has always been part of Western civilization. Cleanliness is a sign of wealth and of good character.

Next we must look at clothing. One thing that people today often assume is that most smell comes from the body when in reality the smell mostly comes from the clothing. Cloth soaks up the oils and the dead skin cells that we produce and, unless washed, keeps those smells. Today we have the added issue that quite a bit of our clothing is made of plastic. Polyester or poly/cotton blends are incredibly common. As polyester doesn't "breath" it tends to keep the smells that rub off on it. In addition, our bodies produce more sweat in polyester clothing as it does not help with temperature regulation. This was less of a problem for people of the past since their clothes were made of natural materials, some of which have anti-microbial properties. The most common cloth in Europe for most of history has been linen and wool. Both of these fabrics have anti-microbial properties that keep them from retaining smell the way that polyester does. That is one thing to keep in mind. The second is the way that people wore clothing. Today we tend to wear our "outer" clothing (shirts, pants, etc.) directly next to our bodies where the oils and smells that we produce rub off on them. People for most of history did not do this. They would wear multiple layers which, first off, made the outer clothing smell much less, second off, created an extra barrier between the smell and the outside world. The inner layer would be changed as often as possible, depending on your mean, specifically to cut down on bad smells. Rich people could change their undershirts (shifts, chemises, smocks, rubahas, etc.) multiple times a day, while those lower down the social order may switch between two shirts a week, one worn, the other in the wash. Historian and reenactor Ruth Goodman (Of the BBC Farm series), writes in her book "How to be a Tudor" about an experiment she did where she followed a cleaning routine from 1534 (Sir Tomas Elyot's The Castel of Helth) where she used a linen cloth to rub herself down every night instead of washing. This, combined with wearing linen undergarments and natural materials let her function without problem in modern society. People didn't notice any horrible smell. She mentions that a colleague tried the opposite, washing himself every day but wearing the same clothes. He apparently stank and looked very dirty as well.

A third thing to keep in mind is the general smell of a mostly rural society. Not bad smells, like garbage heaps, or outhouses, but the general smell of living in nature, where things like grass and trees and yes, livestock, will be quite strong. When you spend most of your time outside, you tend to disperse personal scents a lot faster than you do indoors. There is just more air around you to absorb the smell. In addition, heating and cooking was done, almost exclusively, over a wood fire. Wood fires, though pleasant, have a strong and long lasting smell. This smell tends to overpower all other minor smells and this is something mentioned by both Goodman, once again in the book "How to Be a Tudor" and also by reenactor Pavel Sapozhnikov, who lived as an early medieval Russian for 7 month. The smell of burning wood tends to get into everything, especially woolen outer garments, and acts not unlike a deodorant for body smells. You can experience this yourself if you go camping; your clothing will smell of wood smoke and whatever you cooked for some time.

In conclusion: people would smell more that we do today simply because we a) use products that keep the most obvious smell (armpit stink) at bay; b) change our clothing much more often than our ancestors had the means to. BUT the general surroundings of people meant that other scents were more prominent and masked bodily odour, while the natural fibers and layered fashions of the time encouraged less overall smell retention and production.