Eric Rougier has actually had success shooting 360x180 panos with his iPhone and a fisheye attachmentEric Rougier has actually had success shooting 360x180 panos with his iPhone and a fisheye attachment, but Rougier is a 360x180 shooting phenom who can do these things handheld. :)
Generally speaking, this is not a type of shooting for the faint of heart, or the super-low-budget. The simple fact that you typically want to rotate the camera/lens combination around the no-parallax point to get a good stitch tends to mean specialized equipment.
For a very low budget, you're not going to get fantastic quality. The cellphone + fisheye attachment might work out, or you might consider using a mirrored ball and do the lightprobe thing and creating a pano with the free Ornament PS plugin from Flaming Pear (docs), but it won't equate to doing this kind of work with a dSLR and proper fisheye lens.