How to fix the ugly hiding under the sink.
With basic tools (Hammer, saw and a screwdriver) and under an hour you can go from nasty and smelly to sold and aromatic.
Start by removing the old particle board. I don't have an answer why every cabinet maker chooses particle board to go underneath a sink. It is the absolute worst material for that location. A small leak will make it sponge out and since it is dark and the particle board absorbs moisture it is a perfect mold and mildew host. A hammer works just fine to break it up and tear it out.
I personally prefer to use cedar fence pickets to fill. Cedar is naturally insect and mildew resistant and cedar fence pickets make this projects budget under $15.00. Cut the boards to the width of the inside of the cabinet, after that I use 80 grit sandpaper to clean off the rough cut surface. You can sand the surface until it gets to the smoothness that you prefer using a higher grit sandpaper after the 80 grit if you like.
After sanding the cut planks start placing them inside of the cabinet. A typical cabinet floor is 24.5" from the front to the back and 35" wide. Two six inch six foot tall pickets (using 3 boards) will get you to this point. One four inch fence picket will produce the filler for the remaining seven inches for a perfect fit.
Here is the completed look after using screws to secure the floor in place. It doesn't need a box of screws, one on the left and right of each plank should do. Stain is not required or recommended as the stain would block the insect repellent feature of the cedar. If there is a spill or a small leak (even bleach) it wont destroy the planks (yes, those boards in the middle are wet) and they will dry and be unaffected.