Is there a size limit on what you can take into some museums and also the Vatican? I read about some restrictions on travel bags.
Here's all the details about bag sizes that must be checked at the Vatican: http://mv.vatican.va/3_EN/pages/z-Info/MV_Info_Consigli.html At the Borghese Gallery everything has to be checked.
The complete answer is that there's not an answer. Nobody, not even the Vatican, measures and nobody has an airline-style seizing frame. There is no uniform rule among places nor among door dudes. Some places have storage/checkage for what you can't take in, some don't. Take your chances - - which are better if you go small.
FWIW, some museums and churches won't let you in with a back-pack style bag, but will let you in with an identically sized messenger bag.
I only remember checking my classic Rick Steves' backpack in a few places. The Uffizi, the Borghese, the Scrovengi Chapel in Padova, St Marks Basilica and the Capitoline Museums. Everywhere except the Borghese I was still able to take my purse and it wasn't a big deal anywhere.