An important but often overlooked aspect of maintaining your cut throat razor is storage. Razor blades are made from steel which is susceptible to rust and corrosion so keeping them dry in the humid environment of a bathroom is critical.
The simplest method is to keep the razor wrapped in a towel. This is cheap and easy and will keep the razor dry while still allowing it to be easily accessible.
The main consideration is to keep it out of reach of kids or being bumped and dropped onto the hard, tile floor which could result in damage.
A second option is to keep your razor within a leather pouch. These pouches are very common and provide a protective barrier against scuffing and damage.
Be mindful of excess moisture that can accumulate in difficult to reach areas, particularly at the pivot point. Make sure you air dry the razor completely before putting it into the pouch.
Thiers-Issard leather pouches can be purchased from here to protect your straight razors.
The most extensive storage device for your cut throat razors is a wooden box. This gives a lot more protection than any of the other options as well as giving your setup a classy look/feel.
The Cut Throat Club wooden boxes have a space for your razor, strop, brush, and stropping paste. They also have felt inlay to absorb any additional moisture left after each use. The lid can also be fastened to prevent unwanted attention.
There are also other boxes available which are designed to store 8-10 different razors in a single location. Some people also use cigar boxes to store their razors in as the size/shape of the storage slots is very similar.
If you are planning on storing a razor for an extended period without using it then you should consider oiling the blade to prevent rust and corrosion from occurring.
As a side point - when storing your shaving brushes it is also critical to make sure they are completely dry and excess moisture in the knot. Water in the deep, base of the bristles can damage the glue over time and cause the brush to fall apart.