You can publish them yourself. If you do that, you retain all of the rights to the stories and the book. That way, nobody can get the rights to the book and throw them away.
To publish an ebook, you can make an epub file and distribute it through Kindle, Apple iBooks, Kobo, Nook, Smashwords, and other places. They won't steal it. They want to sell it for you, so they can take their 30%.
To publish a paperback, you can design the cover and the interior, make PDFs of those, and use CreateSpace (print-on-demand) to print and ship it. CreateSpace sells through their own online store, and through Amazon. Depending on how you price the book, they can also distribute it more widely for you (so that it could end up in brick-and-mortar stores).
I do all of the above. There are likely many other options, especially for ebooks.