How can you make it so everyones username can not show in game?

Simple.

In studio in the Explorer (by default on the right of your screen in studio), there is an item that says "StarterPlayer". To be able to edit it's properties, at the top of your Roblox Studio, there is a bar that says "View".

Press View, and within it you will see Explorer, Properties, Game Explorer, etc etc. You want to click Properties to enable you being able to see the properties of each item or group of items you select.

Now with Properties enabled, click StarterPlayer, found directly under StarterPack inside Explorer.

Left click it, and you will see in Properties "Data", and in Data, you will see "Class name", "HealthDisplayDistance", "Name", and "NameDisplayDistance".

To the right of the tag "NameDisplayDistance" there will be a number, which by default is 100. What you want to do is click that 100 and using your keyboard or numpad (if you have one and or prefer it), type in 0.

What changing it to 0 means is that players inside your game will be able to see other players names at 0 studs.

You don't need to know what studs are or why changing it to 0 works, but it just works.

Give it a try, and if it doesn't work make sure you find out what it is and hopefully tell me!

Thanks!

Tomstah and Titaneumcat's answers were very nice, but since you aren't accepting them, I assume you want to disable the playerlist. (or both)

Here's how to disable the playerlist:

LOCAL SCRIPT

local StarterGui = game:GetService('StarterGui')

StarterGui:SetCoreGuiEnabled(Enum.CoreGuiType.Chat, false)

source

There's a property in StarterPlayer called, "NameDisplayDistance." Change that to 0.