It is possible to train at home. Just know that you'll be learning at a much slower rate than a formal student would be learning at, and in the long term, you'll not be able to reach the same standard as them. And it's very possible you could end up with flaws in your fundamentals.
Your best bet for training without formal teaching is to watch as many instructional videos as possible. Buy some bare basics (grappling gloves, boxing gloves, heavy bag, focus mitts/thai pads), used if you're light on cash. Get an interested friend to explore grappling with, and to hit focus mitts/thai pads with. Make a training log here (and/or on other places on the internet) and post some video of your training to it, and hope that some people will give you some constructive criticism. And then get into a gym/school as soon as you can.
If you're looking for help with strength training, that's definitely something that can be learned without mentoring, although it helps. Just do your research, experiment, and be safe.