Simple & Cheap dance floor instead of carpet
The simple approach (and the one I'd suggest above any of the following) is to have a sock-hop (take off your shoes and dance on the carpet) and the most workable approach to a one-time need for a hard floor is to rent a dance floor (they are a standard rental-store item, as seen at weddings done in huge tents, etc...) 8x8 is a rather tiny dance floor for "a large group" and raw plywood in your living room is not going to enhance the decor (rental floors are usually parquet or faux-parquet)

If for some reason you want to buy something you'll need to store, these days laminate flooring (that interlocks at the seams) might be both more practical and cheaper than trying to get plywood to stay flat. It will also store more compactly. If you opt for large sheet goods, tempered hardboard (Masonite®) will be a better bet than plywood (it's far more prone to stay flat) and can be joined with a heavy tape. 1/4" will work better than 1/8"