Yes. The term gymnasium is actually derived form the greek work gymnos. Most greek and roman communities would have a gymnasium where people would engage in athletics such as wrestling (completely in the nude). These structures were large, open air facilities(think of a modern stadium). Over time, gymnasiums became a place of social gathering and health promotion. Many gyms had baths and saunas attached that athletes would go to after their work-out.

In his letters, Lucius Annaeus Seneca (4 BC – AD 65) complains about the distractions of living over a gym:

Beshrew me if I think anything more requisite than silence for a man who secludes himself in order to study! Imagine what a variety of noises reverberates about my ears! I have lodgings right over a bathing establishment. So picture to yourself the assortment of sounds, which are strong enough to make me hate my very powers of hearing! When your strenuous gentleman, for example, is exercising himself by flourishing leaden weights; when he is working hard, or else pretends to be working hard, I can hear him grunt; and whenever he releases his imprisoned breath, I can hear him panting in wheezy and high-pitched tones.

(From Epistle LVI)

So gyms haven't changed much in 2000+ years. :)

I found this question really interesting and I'd like to piggy back off it if you don't mind, OP. The first part may be a question for anthropology, but I'm curious as to what time in history people began dedicating time to physical fitness outside of what they already did for hunting purposes and what not. I'm also wondering when people began exercising purely for aesthetics, and not for functional strength. Were there any examples of ancient Romans lifting just to look good?

The Baths of Caracalla were one of many bath houses built in Ancient Rome, used for fitness and hygiene by Romans, both men and women alike.

The Romans Bathouses often had spa-like services where they would offer hot baths, cold baths, mineral baths, mud baths, and after annointing you with scented oil, would scrape off dead skin and such as well. Understanding that the masses had no access to running water in their homes during the centuries of Roman rule, making these baths important factors in preventing the spread of disease. This final claim is unsourced, but Rome was a city of over a million people around the 1st century AD, and this kind of concentration of population demanded certain hygienic services provided to the populace to avoid the spread of disease by keeping the potential hosts at a certain modicum of cleanliness.

These "spa" services were available to every citizen of Rome, and was one of the more egalitarian services of the time. Rich and poor both tended to share bath times together. Men and Women were both able to access the baths, but on different days.

Regarding the focus of your question, these bath houses also offered athletic training facilities to Roman citizens. Many of these traditions were inherited from Ancient Greece, which also utilized these kinds of bath houses. Typical activities were wrestling, javelin/shot-put/discus style throwing, and weightlifting (although not the typical barbells we see today, more like picking up heavy things and moving them over there).

The actual bath house buildings themselves were double walled grand architectural achievements. Fires were lit beneath the baths to keep water hot and circulating through the walls, helping all of the many functions of the baths. These baths were also decorated with mosaics, depicting possible physical activity for athletes.

Finally, much like gyms of today, these baths served greatly as social meeting places. With large groups of men or women gathering together , it was inevitable that socializing would take place.

You can find evidence of Roman bathhouses as far as england or into eastern Europe, althouhg some are in better condition than others.

Here is an interesting site regarding the importance of Bathhouses in Roman Society, but for the above text most of my information has stemmed from varied sources, which I learned while leading guided tours in the Baths of Caracalla.

Hope this is helpful!