Is talking to a professor about personal issues acceptable? I've been struggling a lot recently with stuff thats been going on, I'm in therapy and receiving help but I'm still having a really hard time. I have a professor who's always asking if I'm doing okay even though I've never told him anything about what's been going on. The other day he came up to me and told me if I needed anything, or even just someone to listen, I could come to him. I kind of want to talk to him, but I'm scared of bothering him and I don't know how I would go about it.

If I should talk to him, should I go to his office hours and ask? Or should I send an email because he doesn't have office hours for another week?
Obviously, how you approach this will depend on what your relationship with the professor is like. That they have personally asked how you're doing is a great sign that you aren't an anonymous face in a large lecture hall. In many ways, having yet another person to talk to---even if it's just a brief session to unload your worries---is hugely beneficial. Pain festers in isolation! I would wager that most any professor has seen the gamut of student depression and other life circumstances. You will not be the first student they've had who has struggled. From their comments, it seems like they suspect something is troubling you and want to check in and see what support they can give.

However, there are some power dynamics at play that are worth acknowledging and considering. 


Particularly in the US, professors are considered mandatory reporters. As such, they are required to notify the appropriate people in some circumstances (e.g., Title IX violations, crimes, imminent self-harm, etc.). Be aware that they may not legally be allowed to keep your conversation confidential. 
Outside of mandatory reporting, know that professors have to strike a balance between being compassionate and caring versus the needs of their other students and university policy. 
Professors will often be aware of resources that you are unfamiliar with on campus---from counseling services, to support groups, to tutoring services. This also cuts both ways: professors are not licensed therapists and cannot supplant the role of one. They are, however, genuine humans with human emotions and compassion.
If outside circumstances are impacting your classroom performance, then this is an excellent reason to reach out to your professors. I would much rather be alerted early, even in vague terms, that there are external factors at work, than learn at the end of the semester after I've assigned grades. It is much easier to work out a proactive plan (such as meeting regularly, building a schedule for catching up, or even filing an incomplete grade) when there is time left to work with. If you think you're falling behind, talk to your professors as soon as possible. Many are very accommodating and receptive. That being said, this should not be considered an expectation and professors must always balance what is fair to the entire class. 
Choosing whether to reach out via email or in person (during office hours or similar) is highly dependent on you. If the idea of talking about your struggles in person is overwhelming, then a vaguely worded email can be sufficient to clue your professor in---never feel like your are forced to share more than you are comfortable with! It is not your professor's job to decide whether your problems are "bad enough." If you just want to let them know that you're struggling and need help, then don't feel pressured to say anything more than that! However, if they seem receptive, even general details of what you're dealing with can help steer the conversation and course of direction. If you're up for it, having a face-to-face meeting can be more productive (again, doubly so where your professor has strongly hinted that you are welcome to come by even if you just need someone to talk to).