I just had a loud and emotional outburst at work, what should I do for damage control?

Not enough information. Were you the one with the outburst, an employee, or a customer? Are you in a position of power or was the person with the outburst?

Depends on what the outburst was about, and who or what it was directed at. Was it about the company, a manager, a co-worker, behavior, policy, something particular that occurred, depends on the circumstances? Political, religious, was there hate speech, racism, bigotry, sexism, misogyny, threats involved, many factors to consider. Also depends on who it was directed at, some you can say & do whatever about and it doesn't matter, others are untouchable & you can't say anything about, point out mistakes or even mention that they did or said something wrong or it is you who will be threatened and attacked by others & management.

apologize on a 1 on 1 basis to those that may have been affected by the outburst

Approach your manager and ask for some time to speak with them for counsel. They may be able to offer advice on how to process times of emotion, in a business environment. This should be a chance for them to mentor you.

Apologize to whoever heard it if it was inappropriate. Have an honest conversation with your manager about it. Reach out to HR and ask their advice. Usually being proactive about it and being honest will go a long way to keeping you in good graces.

Apologise and take responsibility for the incident, and ask for help. Take serious personal action to address the mental health problem at the root of the outburst.

These things happen. Whether in sadness, joy or anger emotions are apart of living life and while it might be unpleasant in the beginning, the best thing to do is nothing. Something else will come along soon enough and replace this outburst as the main topic of office discussion. The only exception would be if you need to involve HR as ideally you don't want to be fired or take a performance hit because you are seen as too emotional.

Excuse to your boss!

Send apology memos to anyone affected, otherwise depends on specifics of outburst, own your mistakes