YES, it is possible. I do it all the time since all but one of my rooms are custom mattresses of odd shapes. Most of my sheets are custom made by yours truly. It takes time (at least for me - I stare at it for a while before cutting anything). Lay the sheet over the bed and use chalk and pins to mark the corners. Sew darts in the corners, trim the excess, then either sew the elastic in the edges or make a casing and feed the elastic through it. I make all our sheets, so-called blankets, coverlets, etc. I sew all of the soft goods here, even upholstry and canvas enclosures.
If you're short on fabric, you can cut a "plate" which is the pattern of the surface of the mattress. Then you can sew on a "side" around it. this means you can piece together the side (it's got a name, I can't remember) - basically, you need to assemble a piece of sheeting that is somewhat deeper than your mattress and sew it to the top plate. Hard to explain without photos. If you're coming up short on fabric, the side panel can be pieced together to make one continuous piece of fabric that is something wider than the depth of your mattress and long enough to equal the circumference of the mattress. Again, hard to put into words...this is why a set of sheets for a yacht can cost several hundred dollars. I chose to sew (because I can) rather than pay.