Is it possible?
Sure! Some houses have had rounded corners from the day they were built. Others have had it added in later because the owner wanted that look.
Well, we don't know your DIY skills, but I'm willing to bet you can! With some practice. Eventually.
To be honest, I've not seen rounded corner bead sold in the stores, but I've not specifically looked. The only thing I've ever seen is designed to have a square corner. However, the preformed corner bead doesn't come to a sharp point, it's got a bit of a rounded edge. It's designed for a fairly sharp corner, though.
As noted in the comments, it seems rounded corner bead is available for sale - it's just that I've never seen it. You will have to deal with some sort of transition at the baseboard molding, if you have baseboards.
To my knowledge, rounded corners was done on walls that were plastered and it takes some skill to get it looking good, but you're free to practice in your own home until you get it right.
Sharp(ish) square corner bead is put up on drywall (or other interior wall surfaces), because putting up the prefab corner bead and mudding it to a "sharp" corner is quick, easy, efficient, and cheap. If you'd like some other corner treatment, you are free to install it yourself, or pay to have someone do it for you.
Unless, of course, you're renting, in which case you'd better check with the landlord first or you risk losing your damage deposit, and maybe being charged to revert back to the square corners if he doesn't like it.