I taught myself to play with Online Bass Lessons at StudyBass.com , I've probably got some bad habits but at least I'm playing and enjoying myself.
Yes you can do it. And with online videos it's easier than ever to teach yourself an instrument. BUT Jabsys is exactly to the point. If you want to learn how to do something the easiest and fastest way is to have someone who already knows how show you! I'm mostly self-taught on bass and I'm here to tell you that you can't help picking up bad habits and it's a BEAR to "unlearn" them later. And of course I DID have study on other instruments (piano, drums) so I wasn't a babe in the wilderness. You didn't say if you already play another instrument or if you are just a rank newbie.
Point is with a good teacher even for few lessons to get all the BASIC stuff down, they can tell your bad habits BEFORE they are habits. They can tell you are doing something wrong AND WHAT IT IS even if their back is to you! That kind of thing will save you untold hours of grief later, believe me. AFTER you get going, then you can drop the lessons. From that point on it's all about practice and that is about your own self-discipline. If you have it, you'll be able to woodshed anything you need without spending anything but your time. When practicing always remember to start SLOWLY. Always play it at a speed you can play PERFECTLY. Speed will come with time. Forcing speed only makes you practice PLAYING IT WRONG. Not a very helpful thing to do!
My next noob suggestion is to find an experienced bass player and talk them into going along as your "expert" when you go to buy your bass. LISTEN to them! Even if you think they are dead wrong, DO WHAT THEY TELL YOU. I assure you, you will not be sorry you did!
And lastly, I'd urge you NOT to be swayed by the small "practice amp" story. Yes they are cheap, but that is because they are worthless! I've got a stack of them here. Good only for driving headphones. Look for a nice used combo bass amp in the 75-150 watt range. Brand does not matter. That is big enough that your bass will actually sound like a bass and you can even use it in small venues and basement jams. I would not worry about the "perfect" bass or "perfect" amp at this point. Wait until you have enough experience that you have a better idea what you want before dropping a lot of cash.
OK, one more than "last". I don't know if you just want to learn bass to make recordings or what, but generally speaking a bass isn't much fun to play alone. Find some friends who play other instruments at about your level of noobness and try to get together to have some fun. Work up simple things you can all play together. It will be inspiring for all of you!