I know I'm in the minority here, but I absolutely believe you can learn a martial art from the Internet, books and videos. You can also learn to fight effectively without any resources at all. The biggest question is should you, not can you. Some of the big factors are, how dedicated you are, how risk tolerant you are, how lucky you are, and how physically talented you are. Those same factors apply to a lesser degree with good instruction too. Now, whether you will be doing some sort of fully authentic traditional art? That's a question for those who judge such things.
Some examples of what I'm talking about:
Let's take a sport art, say boxing, and a group of young people who want to learn to box in someplace without any resources but a garage and what they can get online. Well, they can buy gloves, mouthguards, heavy bags, jump ropes, etc. pretty much anything a boxing gym would have online if they've got the money. If there are a handful of them and they use the roughly $100/month each that they're saving by not paying for instruction it won't take long to have a really nice setup. Then they get a copy of the rules and some videos and books on boxing and they can start boxing. If they stick with it and work hard and are willing to work through or are lucky enough to avoid serious injuries caused by their ignorance they'll eventually achieve some skill at punching and avoiding being punched. Will it be boxing? If they follow the rules of boxing I'd call it boxing.
Will they be as good as someone who got really top notch instruction? Probably not, but they didn't have that available anyway. Will it look like boxing? More or less, probably more. Will it make them better at fighting in a general sense? Almost certainly. Would they have been better off studying TKD or BJJ if there had been a good school for that locally? Probably, but not if they only really wanted to learn to box and wouldn't have stuck with something else.
Another example; Tai Chi. If you live someplace that doesn't have any Tai Chi classes but you really want to learn it can you do it from videos/books? I'd say it depends on your goals and whether or not you've got the patience to do it that way. If your goals are to get the health benefits of Tai Chi, I expect that you can get a lot of them. Things like improved balance and flexibility are likely results of slow controlled movements even if they aren't perfect Tai Chi forms. Improved mental focus and relaxation? Again, I suspect that you don't need to do Tai Chi perfectly to get some of these benefits from doing something Tai Chi like. Will you get the same benefits as if you'd trained with a good instructor? Again, probably not, but unless you really screw things up it's probably a lot better than sitting on the couch.
If you have a training partner you can learn at least some of the lessons of push hands. Much like boxing, if you wanted to learn the competitive form of push hands you could read the rules and practice with other people who had read the rules and eventually you'd be doing push hands, though you might or might not be very good at it. I have no idea how forms competitions work, but I suspect this is an area where learning from video might be harder, but I really don't know. Would some people say that you weren't doing Tai Chi, or that you weren't doing the particular style that you had been studying? I'm sure that some people would say that because I hear people say that when talking about people who have theoretically learned Tai Chi from supposedly qualified instructors.
If you just want to learn to fight and you have a very high tolerance for risk or perhaps enjoy it, there's no need for any martial arts classes, videos or books. Lift some weights, and then start a fight club if you can, or just go get in bar brawls or talk smack to obvious street criminals. You may end up dead or incarcerated, but hey, I expect that prison will give you a lot of opportunities to practice. Being a bit more serious, I grew up in a pretty violent time and place and one of the best fighters I've seen anywhere was a friend's older brother. He'd wrestled in high school but no martial training outside of that and I never saw nor heard of him grappling in any of his street fights. He just liked to hit people and he practiced it a lot. I'm sure that it helped that he was about 6'3", worked in construction and lifted weights, but he was amazingly fast and could hit really hard. He started fighting with anybody he felt was asking for it in junior high when the stakes were lower and just kept doing it and getting better and better.
Now, are any of these examples going to provide you with the same level of safety, efficiency or overall quality of a really good instructor? Probably not, and almost certainly not all three of those things. Will you be able to say you're certified by someone to teach a particular style or have a black belt in something? Not unless your online training program provides that and then it's only worth what people think that's worth, just like any other ranking. But I think you can absolutely develop some real skills without a live instructor if you work hard at it. And let's be honest, with all the talk I hear about McDojo's and fake instructors out there, taking a class is no guarantee of good instruction either.
I've learned a lot of things just from books. Besides the things that people normally associate with book learning, like how to write BASH scripts, I learned a great deal about performance driving from reading a book on how to tune your suspension and then applying what it said about the physics of how suspensions work to how I drove. I'm sure a performance driving course would have been faster but I got pretty good results pretty quickly as a side effect of something else I was trying to learn. I doubt that there's something special about martial arts that make them a unique physical skill that can't be learned from independent study followed by practice.
