Programming isn't a dead end. You can move into management, or if you're happy programming you can still program. If you can't find a job, you can freelance. It's not the type of skill that you need a lot of fancy equipment for (i.e.- you aren't flying planes).
, or do I have a chance of becoming good at programming?"
Being good is subjective. If you want to be good at programming simply reading the right websites, books and learning new things will put you ahead of 50% of the programmers out there. If your idea of good is "Employable as a web developer" you should be fine. If your idea of good is John Carmack, then you're probably not going to end up being "good" by that definition.
Also to most employers, especially ones who don't delver software as their main business function the idea of a good programmer is someone who can deliver on deadlines, adapt to changes in specs, and get along with their coworkers. If you're going to work for a company that makes software as their main business practice, their standards will be higher. Their idea of a good programer is probably someone who has read TAOCP, knows design patters, knows whatever framework is currently trendy and can read the mind of their interviewer and know what books/blogs they like/respect.
Good luck. My dad was a programmer, just as I am. He was laid off when he was in his late 50s, and the only thing that kept getting him jobs were his contacts he built up over his long career. Another piece of advice: Make "friends" who appreciate your skills.