Sounds like you need a buffer. A person who's job it is to keep everyone away from you while that person records what everyone needs and presents it to you at specific times.
Usually when a single person is swamped with questions by co-workers, this will be because the information isn't properly documented in some sort of internal repository.
Wouldn't be the first time I've heard it argued that this takes too much time to implement.
I'd argue that it takes far less time than fielding all those questions manually.
My first question in these situations is: why didn't they go for the internal documentation? And if they did, why isn't that good enough?
I've tried Confluence but without strict adherence to a set of internal rules, that can easily become a mess.
My company originally wanted me to micromanage and direct everything, but I've found that real leadership is when you can give smart engineers autonomy and entrust them to build and execute a plan. I am there when they need me and can jump in as needed if they get in over their heads.
Is this an emergency? Interrupt them. Is this a minor inconvenience? Could this be talked about in the daily tomorrow morning? Is it something that someone else should actually know about? Block the information from the devs and deal with it in the chosen way.
Edit: to answer your question about affording one, it seems around 5 developers can share one secretary, and split the cost. Incidentally, this is a usual setup with a PM as well, but again, they often seem to think their job is to manage the developers, not the project (which they also often don't do).