After searching the web and testing out multiple different techniques I finally settled on a method that gives the following result.
To recreate this effect in your scenes you can follow the following steps:
1. Right click in the project window and click Create - Render Texture
2. Change the size of the render texture. In the example above I used 256 x 128. If you want more, smaller pixels try doubling those numbers, or halving them for the opposite effect.
3. Turn off Anti-Aliasing and switch the filter mode to 'Point'.
5. In your scene camera, set the 'Target Texture' field to your render texture.
6. Make a plane in your scene, somewhere far away from the rest of your scene, and assign the new material to it. The dimensions of the plane should be the same scale as the size of the render texture. For example, the size of my render texture is 256 x 128, which is a scale of 2 : 1, therefore the scale of my plane must also be 2 : 1.
7. Make a new orthographic camera looking this plane. Also be sure that this new camera is the main camera so that it renders to the screen. Adjust the size of the camera so that it matches the size of the plane.
It's as simple as that. You should now be able to see your scene with a pixelated render effect.
This method has a very similar, if not identical result and works for people who do not have access to render textures for some reason or another:
This method allows you to create a pixelated effect in certain areas of the screen rather than over the entire screen, allowing effects like the censor bar in the sims games.