Why is the iPhone 5s battery so awful?

Just got brand new 5s yesterday. Immediately turned off all of the battery-hogging features (LTE, location services, notifications, app refresh, spotlight, airdrop). I don't hav any music or videos on here. All I used was safari and email. But the battery drains before my eyes. What the hell? Even my 4-year old iPhone 4 lasted longer. What gives?

Get it replaced.

That sounds like a faulty battery to me, i have the 5s and my battery life is very good and that is with all of the things you mention turned on.

Apple are very good with customer service generally so the best thing to do would be to return it to them, you will need to make an appointment at the genius bar where they will test it and confirm the fault then they will replace it for you.

Before doing so make sure you have updated to iOS 8, Apple will not deal with the phone unless it has the latest update on it.