Diagnosis
Yes, it is normal for the device to get warm. The main processor generates heat during use, especially streaming 4K HDR content. The aluminum body acts as a heat sink.
How to Fix It
Ensure it has adequate ventilation. Do not plug it into an HDMI port that is recessed or enclosed by the TV's back panel. Use the short HDMI extender cable that came in the box to move the device away from the TV and into open air.
If That Didn't Work
If it's too hot to hold comfortably or shuts down, the power supply may be faulty and overvolting it. Immediately stop using that power adapter. Use only the official 7.5W Google power adapter.
Heads Up
Do not cover the Chromecast with anything or place it on soft surfaces like a couch. This traps heat and can cause permanent damage to the internal NAND storage.