First, how not to be a lumberjack:
Second, and I didn't know this, but lumberjacks don't really exist anymore. No, these days, they are called loggers. The term lumberjack, or lumberjill for women, is reserved for woodcutters prior to the introduction of modern logging equipment like chainsaws. These guys used axes and hand saws, were the manliest of men. In fact, they developed their own culture around their profession embracing strength, masculinity, and the confrontation with danger. In fact, even today, logging is still one of the most dangerous professions there is.
100 years ago, the way to become a lumberjack was to be strong and be willing to do the work. Be willing to risk your life, because that's what it was. The work was migratory, so you had to be willing to move around and live in logging camps, which were often as dangerous as the work. The pay was low, and the work was hard, but there was also a strong feeling of brotherhood and tradition among lumberjacks which, I suppose, made it worth it to those willing to do the work. Also, it was a good place to disappear if you needed to do that, provided you could do the work.
The working conditions aren't so bad anymore, but it's a little more difficult to become a logger these days. You can't just walk up and get hired on because you're strong. Logging companies want people with experience, and you can't take classes for this stuff, so the only way to learn the trade is to do the trade. Small, local tree trimming companies are good places to get the necessary experience, and they are often willing to take unskilled workers and train them up: how to use a chainsaw, how to climb a tree (and be able to use a chainsaw while up in the tree), and how to fell trees in difficult areas. City governments can also be places to get hired on to learn this type of work. If you feel logging is the career for you, it's possible to move on to a real logging company once you've learned the ropes.
If your heart really lies in being a lumberjack, a real lumberjack, there are still ways to do that as the lumberjack culture has been kept alive these past 70 odd years through lumberjack competitions and the like to determine who the real men are. These competitions require the traditional skills of a lumberjack: ax throwing (you know, to take down those trees that are trying to escape), ax chopping, individual and team sawing, log rolling (a favorite in cartoons), and pole climbing. The only real problem with this is that if you really want to be able to win these competitions, you have to be willing to make training for them the equivalent of a job, so you better be independently wealthy or still live at home with your parents.