Exporting Image and Keeping Indexed Colors

"However, when I load the pixelated image onto excel, or img.com it no longer only uses 5 colors. when I zoom in it looks blurry and you can see the different color shades of red."

It seems you have problem with applications in which you are trying to use image(s) and what that apps are doing to your original image.

If you export image from Photoshop and later open again in Photoshop, zoom to 100% and judge. Everything looks fine? Problem is then in apps, not in Photoshop.

»Of the image?«
Of what you are describing but we are not seeing … so the image at View > 100% in Photoshop and the image placed in excel. 

»I have saved as JPEG, PNG, and GIF.«
jpg should be a non-starter due to its lossy compression. 
As for png: png8 or png24?  

»I also tried saving as GIF and "saving for web and devices" choosing 5 selective colors with 0% dithering«
When the image is in Indexed Colors already saving a gif via Save for Web should honor the Color Table by default, as far as I know. 

»when I uploaded to imgur.com it could not be loaded.«
How is that relevant? 

In any case the display of an image in another application (like excel) would naturally depend on what that application does to images. 
So if excel should downsample placed images that would not be an issue with Photoshop. 

How exactly are you exporting? 

Could you please post screenshots with the pertinent Panels (Toolbar, Layers, Channels, Options Bar, …) visible of the original image? 

If you export as indexed color from Photoshop, then the file has indexed color, period. Nothing changes in the file, the numbers are what they are.

Indexed color doesn't support color management. So there is no reference to the colors, and outside Photoshop they will be treated randomly according to a lot of outside factors. Color shifts will happen, and it has nothing to do with Photoshop.

If you need absolutely predictable color, you need proper color management procedure. Color management is the solution to exactly this problem. But then you need to use color managed software only, and you need to have an embedded color profile in the file (which indexed color doesn't support).