It is possible to become a miner, I mine out of my own home myself and use the heat to warm where I live. Though it does take a significant investment, and you need a good electricity rate and you can not just use a normal computer to mine profitably, you need highly specialized hardware. Over time this situation might improve with the commodification of ASIC's, however today it is certainly not for everyone.
@freetrader This is not necessarily bad news I would say, in Bitcoin this comes with certain pros and cons, it does make Bitcoin more secure for all of the users of Bitcoin, so that is good news at least.
@VeritasSapere you are right of course, it is still possible through so-called "pool mining" if you have powerful enough hardware.
One can buy ASIC-based mining equipment like an Antminer S7 from companies like BITMAIN, for example. But that is going to cost many thousands of dollars for even just one mining appliance, and even then your mining power is tiny compared to the big farms (so you need to pool together with others to have a good chance to get a payout in reasonable time).
I guess I said "bad news" because many people, when hearing about Bitcoin mining, are hopeful that they can easily participate and get "hands on". But there is that investment hurdle at the moment.
Like you said, with commodification of ASICs, maybe soon they will be more affordable to everyone again - and I can install one to heat the water in my home too!
In the meantime, there are other good ways that a person can help the Bitcoin network.
One is to run a so-called "full node" - which is a complete copy of the Bitcoin software - on your own computer. While that does not mine Bitcoin (usually), it helps to secure the network a little, and helps one gain a much better understanding of how things work.
Another way to gain Bitcoin is to offer some real-world services for it. One could offer a small discount for people to pay in Bitcoin.
Everyone can set up a wallet and receive some Bitcoin - all that is needed is an Internet connection to download a small piece of software.
Internet sites like Reddit also enable people to give each other tips in Bitcoin (through changetip.com).
If you're just starting out you're a little too late to mine it on your own without significant investment. As @freetrader said, it might be a good idea to provide some kind of services for Bitcoin. You will be surprised on how popular this type of exchange is right now (and growing!).
You may want to start by mining other coins. Ethereum seems to be pretty hot now and you can then trade it for BTC if you wish. Be prepared to invest in a graphics card with at least 3 gigs of memory because the DAP size is so big now my nvidia 690 just doesn't cut the mustard anymore. She served me will with doge though. 163,000 coins mined! Back in the day I traded half of them to buy a video game that I didn't even play because someone stole it. Basically, I've got 98k left which is $200-$300 at any given time. If I had not bought that game, I'd have a lot more. I'm never cashing out again until I'm absolutely ready and not for some random thing.