I remember back then when I was younger and had to spend a week at an Uncle's house, I was told by my Dad that he loves to see kids studying really hard, in my quest to impress him I decided to stick to my books once I stepped foot into his house, but the truth is that I was not studying anything (just staring at it), how stupid! I now wonder why I even wanted to impress him. I didn't let myself have fun, neither did I enjoy the book that was in front of me.
I believe there is an enormous difference between being productive and looking productive. Most of the time, we focus on looking it rather than actually being it (well, maybe we wanna impress?). We might refuse to hang out with friends, we refuse to let ourselves relax or do anything that might look unproductive, but at the end of the day we just end up looking it, not being it.
The most important thing is being able to strike a balance between doing the hard work and having fun, you will get the job done, be far more productive and also enjoy life as well. On the contrary, when trying to look productive there is a high chance of not being productive at all (or at least not as much as we want to), because the focus is on "looking it" not "being it".
I had classmates back then that engaged in almost no fun at all, and studied all day, they looked really productive but the results at the end of the day showed otherwise, maybe they really did study hard, but they were not able to strike a balance, have fun and refresh their minds.
Yes, it is possible to be productive and enjoy life to the fullest. Allowing yourself have fun will replenish your mental energy, help you de-stress and enhance your performance.
Don't focus your energy trying to look productive, be productive, have fun, you might appear to be unproductive but let the results speak for it.