Yes, Kodak went through bankruptcy a few years ago, chopped a bunch of stuff up to sell off, and made a complete exit from a few markets. However they are still producing film and related products, and dealing with commercial services.
If you want to work with black and white film then dealing with chemistry is easy enough at home. A changing bag, paterson daylight tank, and a simply setup for chemistry is all you need to shoot and develop for yourself. You can work with colour film, but it is a little more tricky and needs tighter quality control to look decent.
Also if you do want to shoot film these days, then I would suggest considering medium format or larger. The added cost in film isn't that much in medium format, especially if you're shooting a 6x4.5.
I've been shooting a bit of film after getting an old TLR camera a few months ago, and have been enjoying working with it and what it produces. Settling for scanning for the time being, but I'm storing the negatives and will have a body for material to work with when I can get a small darkroom setup going.
Canon EOS 7D | EF 28 f/1.8 | EF 85 f/1.8 | EF 70-200 f/4L | EF-S 17-55 | Sigma 150-500