The subject of this post is a much-contended point amongst thespians. I favor the school of "yes, everybody can give it a go." But mostly because those who lean the other way tend to be a bit extreme. I'll here hit the middle ground with, "anyone can act, but not everyone can act well."
As 2010 Runner-Up Best Acting Coach in New York, Jason Bennett, said on that "The Everything Film Show" thing I keep telling people to check out, everybody needs to find their own process. Most people will never have the chops to be the next Denzel or DeNiro, but many people can find additional possibilities within themselves with a strong dose of humility, bravery, and effort.
To be an actor, one has to be willing to embarrass themselves - a lot. Because at the end of the day, the resulting visual is largely out of your control. You have a Director of Photographer (often with multiple cameras coming at you from different angles) catching you in motion, a Director who's framing the overall scene, and about 800 other things going on in terms of working with others on set. Once all that's done, your image is in the hands of an Editor, who can completely make or break a project - and might just opt to leave you on the cutting room floor. Hopefully, if those in the seats that steer know what they're doing, all turns out well. If not, it's your face on screen either way. It's good to be ready to deal with that.
The raw truth is, anyone can try their hand at acting. There are open calls posted in newspapers all the time, and you sometimes even read of leads getting discovered at diners and such through no effort of their own. (For all of the starry-eyed folks, please understand that the chances of getting struck by lightning are statistically far greater.) But there are calls everyday seeking regular people to fill in scenes.
You can always try out or submit an email application when you find opportunities. See a posting? Tell a friend where you're going and check it out. Notice something interesting in the casting sections of Craigslist? Same as above. Mandy is another great, free site that you can use. There are also paid-usage sites available, some of which I patronize, but I'm saving that for another post because, in my honest opinion, you shouldn't really even be considering them until you are already getting work.
Whatever you decide to do, always be skeptical and exercise caution. My first audition, following a brief and relatively unsuccessful stint in modeling, ended up being with an absolute sweetheart with whom I am still in touch with today. My Mom literally accompanied me to the audition (the only time she ever did so), given my age at the time. That all worked out really well, but you really can't be too safe.
This is becoming a long post, but I would also be a jerk not to mention - if you think getting on set as an extra will be your trip to bigger things, please reconsider. There are tons of hard-working actors who have years of experience doing paid, credited, speaking gigs, who also do extra work fairly regularly to keep bills paid (hellooooo, Central Casting!). Some of these same full-time acting folks will resent those without full-time professional aspirations coming on set simply because you are competing for the same work that is keeping their rent paid. It's good to always be mindful and respectful of that fact, and perhaps a point to consider, if you're still sitting on the fence, that perhaps you should look away. The field is very competitive, and when there is real money involved, it becomes much more so quickly. If you're really not sure about whether you really want to do this, I would humbly recommend shooting for unpaid work just for the experience to give those working full-time as actors additional opportunities to keep doing what they love and keep the roof over their heads. Many people who are borderline curious will find out that they simply do not like being on set.
So, back to the point I got started with - literally anybody can be an extra on set if you want to try your hand at acting. In fact, I imagine it's even easier than ever to do so given the number of items that are today posted on the Internet. Every time I'm on set I seem to meet at least one full-time business professional, student or retiree who had always had an interest and just wanted to check things out. If that somebody on the next set happens to be you, be safe and have fun! It is definitely a unique experience.