It depends.
I think most of the time, actually (surprisingly!), I can.
I am a sensitive person, but I'm also able to compose myself & do what I need to do when I need to do it.
I'm a stay-at-home mom now, but, when I worked, I was 100% there & was able to put personal things aside. However, I admit - the reverse was harder for me, & I'd often come home w/ work and/or come home w/ my frustrations & anxieties around work.
Now, even though I'm no longer employed, if I'm working on a project or doing something related to school or one of my volunteer positions, I'm still able to put aside personal and/or home issues & focus on what I'm doing, if that makes sense.
I spoke at my grandfather's funeral & did so by telling myself as I walked up to the podium, "Right now, you're a speaker telling everyone about your grandfather. You're not a granddaughter right now - You're the speaker."
The day of my father-in-law's visitation & 2 days after he passed away, I taught my class at our co-op because I knew they were depending on me & it'd be hard to get a sub. And, for that morning, I just put my father-in-law's passing & all the associated family stuff to the side & taught the class.
Years ago, I played volleyball in high school. Our coach was also the high school math teacher, & she was a sort of mentor for me. She lived a very disciplined, organized, no nonsense kind of life & had a certain standard of behavior she expected from herself & her students. I remember once, towards the end of a very high-stakes volleyball game, during our last time-out, she told the team, "No matter what happens. We are not emotional bubbles" meaning, if we won, we'd be gracious winners &, if we lost, we'd be gracious losers & conduct ourselves properly.
Over the years, that's stuck w/ me.
I am a worrier & over-analyze frequently, & I also have anxiety. But there's definitely an "on" me who can act & do accordingly & then the person I am when I'm not "on".