We don't become emotionally indestructible robots the moment that we walk into the office every morning. We all "bring our problems to work" one way or another. The question is whether we're able to discuss them with our colleagues and leaders.
In any workplace, there should be a focus on building an engaged and high-performing team of individuals who understand each other and support each other. When two people solve a problem together, their bond becomes stronger. In my view, this problem doesn't necessarily need to be work related either, I see taking an interest in each other on a personal level as only having a positive impact.
If you have worries outside of work, it's inevitable that your productivity will fall, but if there is a degree of trust between co-workers, leaders can often use their experience to help alleviate these concerns. I know myself that if something is playing on my mind it can occupy valuable thought processes. Instead of creatively solving a work problem my mind quickly wanders back to any niggling issues. At the most basic level, leaders can make themselves available to listen and offer sympathy.
Loyalty breeds loyalty - not only in individuals, but in wider teams. When we do everything we can to show our people that we are there for them, people are happy to raise even minor issues. Helping to take care of background concerns by talking them through is far better than dealing with the productivity loss of a distracted mind.
It gives your internal dialogue a boost when you have helped someone out on a personal level. It is one thing explaining to them how to compile a report, but another thing entirely giving them a shoulder to cry on after a relative has passed away.
When you ask people to bring their whole selves to work, you simply have to give them the freedom to express their feelings. This might take up 5% of the working day, but if everyone is far more productive and engaged for the other 95%, then it is more than worth it. You can't calculate the impact of a supportive culture on a spreadsheet, but you can feel it when your collective backs are against the wall. When you already have a culture of supporting each other, hard times are handled with ease.
I suppose it comes down to whether you care enough for others. Halfheartedly asking "are you okay?" is one thing, but sitting down, pausing and waiting for the long answer is another thing entirely. If people recognize that you are genuinely interested in them, they will open up to you. If you are "listening" because you should, then you won't be much help.
When was the last time you really listened to someone? About nothing related to work? Do you think that it made a difference to them?