Showdown doesn't have a built-in function that allows you to play unrated battles in any tier of your choosing. Smogon's main tiers like OU and UU only have one ladder, where the battle results will be scaled on your ELO, GXE and other measures. There exists no ladder for any game that allows you to play unrated, except Random Battles. You can play those without ratings by selecting "Unrated Random Battle" from the format menu, as seen here.
However, Showdown's account system and (non-existent) policies regarding them mean you can basically replicate the effect of playing without results. Just sign out of your account and make a new one, and play battles on that account. The battles will be scaled, but your main account's rankings won't be affected. There's a strong culture for this on Showdown; plenty of people like to play on "testing alts" before they're confident in a team for laddering. The main difference here compared to a truly random system is that you'll always play against people with a similar ranking to your current account -- though the unranked ladder wouldn't be very competitive in the first place. Anyone who is serious would play ranked games instead.
Basically, there's no way to get a truly random opponent (ie. completely ignoring rating parameters and prescribed matchups), but you can get pretty close by making a new account and playing on that one.
