Oh he's a sweetheart! If you want to keep him as a pet, there's nothing wrong with that!
Every cat I've ever "owned" is a former stray that found me! Its probably safer for him to be kept inside (especially if you havn't neutered him, as he can get into fights, contract diseases, and also impregnate female cats, contributing to the overpopulation) and at the same time, he's safe from cars, mean neighbours and other outside dangers. It might take some time for him to really warm up to you, but take it slow, and at a pace he shows he's comfortable with, and you should have a lovely new pet cat on your hands! Congratulations!
Remember to get him into a vet as soon as you can for a checkup, shots etc, and depending on his age, to schedule a neuter
Oh, and just a side note, remember to keep fresh water out for him at all times, not milk. Most cats are actually lactose intolerant!