Yes, and yes...to a certain extent. You can do both, you'll just be more limited in what you can play and how well you can play it.
I originally started playing classical on a steel-string acoustic (self-taught), and continued to play on it for 7 months until I scraped together the money for a classical guitar. There were definitely some disadvantages to playing on a steel-string--higher tension strings made some techniques harder, closer strings also made fretting/plucking more difficult, instrument was too big to play upright in classical position, sound wasn't ideal for classical.
That said, I managed ok on my steel-string, though keep in mind I was only playing beginning/very easy intermediate music. Not sure how well it would work for more advanced music, though I suspect the more advanced the music, the more problematic a steel-string will be. If you just want to give classical a try on your steel-string to see if you like it, I'd say go for it. However, if you do end up really liking it, I'd recommend trying to save up for an inexpensive classical guitar, if you can.
As far as being self-taught...you can definitely do it, but there are lot of disadvantages: Learning stuff wrong, missing fundamentals because you don't know they exist or that you need to learn them, learning more slowly than if you'd had a teacher, having to experiment to figure stuff out, playing inappropriate repertoire because you don't know your level yet, challenging yourself too much or sometimes not challenging yourself enough... The list goes on. I know, as I'm self-taught.
Is it best to have a teacher? Sure, but not everyone has the resources/opportunity to do so. So if you want to learn, I'd say learn, even if you have to teach yourself. You won't be perfect, but then, you don't need to be if you just want to play at home for fun.