I know in Biblical days, angels were often mentioned, but does modern-day Judaism still hold that angels exist, and what do they say angels do? Are they messengers or guardians?
Judaism does have a well-developed angelology (and demonology, for that matter) but it's really more in the realm of what one might call "folk Judaism". In all my years I have never heard a rabbi or anyone else refer to either angels or demons in today's world.
Throughout the Old Testament, which, is directly based on Judaism, there references to an Angel of the Lord. So, in answer to your question, it is a resounding yes.
Yes, they believe in angels...
And also the archangels i.e. Gabriel, Michael.
As far as I'm aware, Jews and Christians both believe the same about angels' "jobs".
Messengers from God to man, and spiritual warriors against demonic beings and fallen angels.