Can a gym be built to supply electricity to homes?
The maximum continuous power that can be generated for an hour by a fairly fit person on an efficient machine like an exercise bike or rowing machine is $\sim 200$ W (olympic-standard track cyclists might manage 400 W).
Let's say that a gym is occupied at any time by 10 people who are doing this kind of intense exercise. Then you might just be producing enough electricity to boil a kettle (kettles are 2-3 kW) and keep the lights on in the gym. Unfortunately there are another 10 people in the showers who have just consumed more electricity than they generated (typical electric shower consumes 8kW, so a 2 minute shower needs 960 kJ = 200W $\times$ 80 minutes).
Does that answer the question?
However, it might be an interesting gimmick to allow people to charge up their phones or other personal devices using the electricity that they personally generate. That would probably be feasible with the right adaptors and transformers.
Yes, these exist. There's one here in Portland, OR, USA called The Green Microgym. But it doesn't generate much energy. People wildly overestimate how much energy a human can produce.
They claim to "have generated 20% of our own electricity" but it is done by "combining human and solar power". They claim to use "Energy-producing cardio equipment (ellipticals and stationary bikes)". I don't have any numbers for how much of that 20% is human power and how much is solar, but Rob Jeffries already covered what a human is capable of: about 200 W.
If all 150 members exercised a very generous 1 hour a day producing 200 Wh each, and all that energy was turned into electricity with no loses, that's a maximum energy output of 30 kWh. To put that in perspective, a refrigerator uses about 1.5 kWh per day. So 150 people exercising an hour a day at peak capacity with perfect conversion to electricity can power about 20 fridges or about 0.133 fridges per person. That's an ideal scenario.
Most of the gym's efforts go into reducing electricity use and waste rather than generation claiming to have reduced their electricity use by 85% compared to normal gyms (per square foot).
There have been claims of generating electricity using human power before. This is entirely possible, but they generally wildly overstate how much power can be generated.
The most recent I've seen is the claim that "60 Minutes On This Bicycle Can Power Your Home For 24 Hours". This is nonsense. Note the total lack of details about the device in the article or video, they only talk about its potential.
As Rob said, the average person will put out about 200 Wh in 60 minutes. A single LED bulb uses about 10 W or 240 Wh per day.
So, maybe if you live in a shack with one light the claim is true. More realistically, it could be used to charge a phone or laptop or power a radio.
As answered above no not electricity, but usable force has been done.
Treadmills originated as a corn mill powered by prisoners. They would stand on a large cylinder and turn it grinding corn.