I know that it's a horrible thing to do. It was a professor at a prestigious University who expressed interest in my ideas. I never responded because severe conflict broke out between me and my family regarding my career choice. And that was a year ago. Conflict is still there but I'm stronger, and I sent a sincere apology to the professor, but I still want to know how bad my situation is if I want to reapply to the same University for a Masters (i don't even dare to hope to work with the same professor).
If it is a coursework masters, then you would have minimal or no interaction with the previous professor, unless the professor is teaching a subject in the Masters. If it is a research masters and you are in the same department, then you would be working or at least bumping into the prof at departmental meetings and events.
Seems unlikely that there would be great animosity. Your circumstances seems very reasonable and most people would not react adversely or take it too personally. I am sure if you are in the professor's department, then you can have a casual chat and smooth over any frustrations. The prof might be able to collaborate or support you in some way as they were previously interested.
It may take time until a student is ready to actually start a thesis. Unless you promised something firmly and flaked out, I do not think that there is much personal harm done.
Of course, the professor may have become interested in other things in the meantime, and his original interest in your question could have evaporated. But it's worth getting back in contact to see how things stand.
While it depends on personalities, you might find that all avenues are still open to you. Professors, even quite prestigious ones, are still people, and most of them (us?) care about students and understand their difficulties.
But, the best way to remove issues from the past is to do so in person. A letter or email isn't enough. Nor can you evaluate the response if you just send a mail. Go to visit the professor and apologize for what happened earlier, but also explaining that you had personal/family issues that were affecting your ability to work effectively toward your future. You can remind him/her of your old work and ask if there is any possibility to continue that, or something similar, in the future.
You might be told no, of course, but if you don't ask then there is no hope. But I think that you would get a strongly negative response only if the person isn't someone you should really work with anyway.