Covering damaged wood cabinet under sink
If I understand you correctly, you are referring to the cabinet floor.  A water damaged cabinet floor is very common.  Covering the damaged floor with laminate or a vinyl contact paper is often done to hide the damage and help protect from further damage.  However, if you are planning on installing a roll out trash can, the floor will need to be solid and level.  Depending on the type of material used to make the cabinet floor and the amount of damage, it may not be solid enough to support a roll out.  Additionally, it may be solid, a good grade of plywood for example, but the long term exposure to moisture may have bowed the floor and this will not allow the roll out to be installed correctly.  So you can cover it with something like that if it is a good quality material that has not rotted or bowed.

If the floor is not solid, you will need to replace it.  This is also a fairly common procedure and it is within the range of most DIYers.  You will need to cut out the old floor.  Once the floor is cut out, use some 1x2 stock lumber to install a cleat around the perimeter of the cabinet lining it up so that the top surface of the cleat is where the bottom of the old floor used to be.  Then cut a piece of good quality plywood of similar thickness to your old floor to the same dimension as the inside of the base cabinet.  You may have to drill for any plumbing penetrations and be prepared to disconnect and re-connect those plumbing lines.  Either finish the plywood with a good paint or stain/clearcoat.                                                                                          You can also use the vinyl flooring or vinyl contact paper here to protect the plywood.  Then just lay the plywood on top of the cleats ans screw it down.  Now you will have a stable base for you roll out trash can.

Good Luck with your project!