The amount of space you need will depend on the kind of lifting you plan on doing or what kind of equipment you prefer.
Here are a few possibilities of what you can do with some small square footage…
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With a Power Rack and Bench Setup
A power rack is about 4ft wide, or it is if you only measure the uprights. It might be wider overall if it has something extending out like large feet or plate storage pegs.
The 4ft figure is pretty universal, because an olympic barbell is around 51-52″ long between the collars to fit over it with some room for slop.
The overall length of a typical barbell is 7ft long, and it will need a free range of movement, and you’ll need room to load the plates on it.
In addition, will you have a bench that you will slide in for bench pressing?
To give yourself extra room to stand on either side and load the 7ft bar, 10ft of space is best. That only gives you 16″ of space on either side to stand. A little less can work in a pinch. It’s going to be awkward loading 45lb plates on without enough room to stand comfortably on either end of the bar to load weights.
Another way to mitigate the situation is with a special 6ft bar that is made with the same inside length as a 7ft bar and with sleeves 6″ shorter on each side. Take care not to get a women’s olympic lifting bar that has the same shaft and sleeve length as that but with a thinner 25mm shaft that most male users won’t like.
As far as the depth of the area from front to back, you will ideally want enough room to slide your bench in and out. Add together the length of your bench (4-5ft) plus the depth of your rack (4-6ft) and the overall space you need is anywhere from 8-11ft. You can do it with less, if you can slide or roll your bench out at an angle.
Flat benches will be closer to 4ft long, because there isn’t as much to them. Adjustable benches might be closer to 5ft long, because they sometimes have space for leg or preacher attachments.
Some basic power racks are only 4ft (48″) deep. The ever-popular Rogue R-3 floor-bolted rack is only 34″ deep.
Extra space in front of the rack will be useful also for things like deadlifts, cleans and overhead presses, some or all of which you won’t be able to do inside your rack.
Cleans are such a dynamic movement that will have a varying range of motion if your form isn’t perfect, and it isn’t smart to do them inside anything but a deep rack, at the risk of hitting the uprights or bar catches.
Deadlifts are nice to do outside the rack too. Deadlifts can be done inside some racks by removing the safety bars. Other racks have safety bars that can only slide up and down, so you can’t get them low enough for your plates to touch the floor, but you should be able to slide the safety bars really high out of the way instead and slip the bar underneath.
Overhead presses can be done inside some commercial racks that are very tall, but most home and light commercial racks are only about 7ft tall, to fit under 8ft ceilings with room to spare, meaning an average height male will likely bang against the bar against the top crossmembers, if not hit the plates on the ceiling. It depends on how tall you are and how many inches of added flooring, such as plywood and rubber, you have under your workout area.
Weight Plate Storage
Don’t forget about where you’ll be storing your weight plates. If your power rack doesn’t come with plate storage pegs, you need a separate weight tree and enough room around it to pull off plates. You can’t just set the plate tree right next to your power rack either, or it will be in the way of the range of motion of the barbell.
So all that being said, just for a basic power rack and a flat or adjustable bench to use in it, you want at least a 10ft x 7ft area.
With a Squat Stand and Bench
A full-sized half rack (also called a squat stand), sort of like a power rack with the front side open, doesn’t really save on floor space. It needs the same footprint as a power rack for stability.
It could work better for you when your area is tight and you need to maneuver around it, stepping over the feet, and you can push the rear of the rack all the way up against the wall and still have room to work because of the way the uprights are closer to the center, unlike a power rack that may need to be pulled away from the wall by 12″ or so to give you room to work. So while the footprint itself is about the same, functionally it could work better.
See this example.