Beeping coming from INSIDE the computer, but everything is fine

This problem is known colloquially as, “No, it’s not coming from the speakers. The box is literally beeping at me.” 

It’s easy to forget that desktop PCs have a speaker inside the case. It beeps once every time the PC is powered on as part of the Power-On Self-Test (POST). I mean, who actually turns off their desktops?
Aside from that friendly chirp announcing a successful boot, the onboard speaker will lurk silently in the inaccessible depths of the computer casing, until that one day when it has something urgent to say. 

When insistent beeping starts coming from inside the computer while it’s running normally, it is often accompanied by appropriately alarmed end users — which is good, because the main purpose of the built-in speaker is to alert nearby humans in the most annoying way possible that the CPU is running too hot. 

It’s telling you, “Hey! I’m dying here. Can you please slow down, or I don’t know, open a window?”
It might not be catastrophic if you fix it immediately, so after you’ve explained that the box is not beeping because it’s about to explode (which is technically true), take these steps:

• Enter the BIOS and note the CPU temperature. If it’s over 140 degrees Fahrenheit (60 degrees Celsius) then it’s getting too hot for comfort.
• Ensure the CPU hasn’t been overclocked by an enthusiastic end user.
• Check that the computer case is well-ventilated and all fans are spinning.
• Blow out the dust on the motherboard and system fans. Do this outside!

If the computer isn’t booting at all and emits only a short series of beeps, it’s time to break out the BIOS beep error codes for the motherboard. This will tell you which piece of hardware is failing, and how.  

Or you can just Google it.