Is it practical to refinish a small section of floor in my rented house's kitchen, and if so, how do I do it?

First you have to decide what kind of flooring you're going to put down. If you just want to do rolled vinyl (linoleum), you would probably have to take up the old flooring first, especially if it's not smooth and flat. Bumps in the lower layer would show through the new flooring. There are also locking laminate floors that don't need to have the old floor taken up first. You just put down a layer of plastic foam sheeting as an underlayment and put the laminate tiles together on top of that.

Sometimes, this can be more cost effective, even though the tiles are more expensive than the linoleum, because of the extra labor involved in removing the old flooring. But, you will need to install a reducer where the new floor contacts the existing floors in room transitions, and put down a quarter round molding around the edges. You should also have a compound miter saw to do the cutting, which can be difficult for a beginner.

If you have to remove the existing flooring from a concrete slab, you have to use a floor scraper that has a long razor sharp blade attachment at the end of a 5-foot handle. This can be a lot of work, especially if there are multiple layers of old flooring on the floor. If you have a wood sub floor, you just have to use a claw hammer or pry bar to remove the old flooring. When I remodeled my kitchen and removed the old flooring, I literally found 7 layers of old flooring under the top layer (the house was built in 1946). But, it peeled up pretty easily. The bottom layer was thin plywood, so I just had to bust that out in large pieces.

Actually installing the rolled vinyl isn't difficult. But, getting it exactly right can be if you've never done it before. Ideally, you should cut the sheet out in one piece, so you don't have a seam anywhere. If the sheet is short enough to require a seam, this can be difficult to cut exactly straight. So, you should butt precut pieces up to each other to form the seam instead of trying to cut a straight edge.

Also, ideally, you should have about a 50 to 100 pound roller for rolling the floor flat (removing air bubbles or glue pockets) after installing it. And, make sure you get the right kind of glue (for linoleum instead of wood floors). I got some glue for a new linoleum bathroom floor that turned out to be for wood floors (it looked almost identical to the linoleum glue can and it was next to the linoleum flooring at the Home Depot). The wood glue never completely sets, while the linoleum glue dries completely when cured.

Another option is peel and stick linoleum tiles. This is the easiest to install. However, consider that moisture may eventually cause the edges and corners to peel up. So, it's not really recommended for kitchen or bathroom floors.

First get permission from your landlord to replace/repair floor.

Second decide what type of flooring you want to put down.

Then go to your local Home Depot store or Lowe's store or both find out when the course you want to take will be taught and take their course in "How To install"

These courses are free. Also this is how my wife and I learned how to install tile flooring in our bath and mud room.

Do NOT put wood laminate flooring (floating floor) in a kitchen. Water has a way of getting under the flooring and it will have to be replaced.

You might also ask the landlord to buy the materials and you furnish the labor to install. This would be a win - win situation for both. I am also a landlord.