I wan tto do something with my linoleum floor beneath the washer and dryer. It tears when we have to put the dryer back into the small area after a maintenance. The washer sits in a metal "Pan" that has a drain under it in case the washer leaks. What can I do with this small area's floor to make it more durable?

Why not install a simple tile floor in this area? You could have the tile person water proof the floor area, install a drain in the floor. Then install a small saddle in front of both the washer and dryer with tile floor trim. Once properly grouted and sealed any leaks would drain out in the floor drain rather then spill into the room.