There is no standard size, and there isn't "personal item" storage on many attractions. For example, on Expedition Everest, you'll find an elasticized net pouch in front of you -- on Mission:Space, there's a hard-sided box -- on Soarin', an underseat pouch -- on Space Mountain, nothing.
That being said, you should have no trouble carrying a small backpack or other bag with you on attractions. My DH always carries a full-sized backpack (containing sunglasses, extra sunscreen, a hoodie, etc.) and I carry a medium-sized crossbody purse (e.g., Vera Bradley "Hipster"). When boarding an attraction that doesn't have a suitably-large pouch or container for personal belongings, DH simply sets the backpack on the floor and places one foot between the straps (like
@CaliforniaDisneyMom described), or, if the restraint system allows, turns it around and wears it over his chest. I typically just leave the purse slung over my body, and tuck it to the front or side, as appropriate. We've never had a problem finding room for our things, or had anything fly around.
If we're going on Splash Mountain, we do make sure our electronics and phones are in ziploc baggies inside our backpack/purse, and on Kali River Rapids, we enclose the backpack or purse inside a plastic shopping bag before placing them in the covered (but often wet) storage area in the ride vehicle, or else wear them on our bodies with ponchos over ourselves.
