Chromecast Audio is a good gadget to cast your music/audio to a speaker via 3.5mm. But I can't find out if it is possible to output to a normal earphone.
Can I just plug in a normal headphone to Chromecast Audio?
There's very little practical difference between a speaker and headphone out at a very basic level (Yes, I'm aware of line level out vs amplified outputs and so on). If you're doing more serious audio, maybe. http://liliputing.com/2015/11/chromecast-audio-review.html does seem to suggest it works though - they tested with a Sony MDR-V6. You might have issues with unusual low or high impedience headphones, and the lack of volume control might be annoying, but it should work.
Works extremely well for me; I route the 3.5mm analog-converted audio directly to earbuds and a variety of headphones. The sound is superior without exception empowering my already capable earbuds and headphones to another level of listening experience.
If you are interested in my recommendations of headphones that won't break the bank, assuming you don't get the sound improvement you are looking for, contact me directly and I'll advise based on your listening preferences - genre, volume, isolation, etc..