Cut the linoleum scrap the size of area rug you want. Often large home stores sell remnant pieces very inexpensively. Once you get your piece home you may need to unroll it and allow it to sit in your environment for one or two days so the linoleum will relax and lay flat.
Place the linoleum face down on plastic sheeting and prime the back side of the linoleum using a paint roller. Allow four to six hours for the primer to dry.
Lay out the pattern on the linoleum using free-form painting of acrylics onto the linoleum surface or by taping painter's tape and using stencils to draw or paint through the stencils. The linoleum can be painted similar to any artist canvas from simple to complex. Often the center of a rug will include a field of color and a rug-style border. Paint with artist-type acrylic paints and brushes. Allow the paints to dry for 24 hours.
Seal the rug using a nonyellowing clear finish. Krylon has some spray finishes that work well over acrylic that are available from mat to glossy. Apply several coats to make the rug more durable.
Tape the edges with a fabric tape that wraps from front to back. Consider spraying the bottom (slippery side) of the linoleum with a rubber backer to make it skid resistant.