Is it appropriate to email an eminent researcher in your field?
Here are some of my strayed thoughts.

Think from return of investments, of yours and your idol's

If you ever perceived that your e-mail would be a "waste of their time," then why send it? I feel that most eminent researchers have a trait of "ignore everyone and head for their goal;" getting acknowledgement and acceptance is probably not their primary concern. A specific e-mail describing how their work has inspired your study/project is probably fine, but I wouldn't go so far to expect they would reply and give specific comments on your hypotheses.

From your point of view, instead of using the energy and bandwidth to send the e-mails, there are a lot more you can do:

There are many ways to show your appreciation

First, they would probably like to see their work being formally cited and, more importantly, applied to the field or crossed into other fields. Each idea geminated from their work is an appreciation by itself, and in the mean time you can also enhance your publication and research paradigm. The plus is: if you have done enough of it, the big shot may actually contact you and give comments.

Second, you can help preaching the researcher's ideas and agenda. You can write blogs, answer other people questions, use their works in your journal clubs or lectures, etc. to subtly introduce the researcher's teaching to the public. Better yet, refine the researcher's ideas, and incorporate into yours. Become a spiritual successor with your own unique approach. And let your career be inspired by the researcher.

You can learn from someone without establishing communication

For some more senior researchers, look for their auto-biography, biography, interviews, and documentaries that feature them. I will probably never be able to talk to Itzhak Perlman, but I learned a lot about him through books, websites, documentaries, and musics that he plays. (And actually, he has a Youtube channel as well, but I am suffering from too much fanboy shyness to write any comment.)

For younger researchers, try look for their blogs, Youtube channels, open courses, or even biographies of their mentors. All these may help you become more familiar with them.

Another way is to indirectly know them. Most of these researchers would have a lot of students or proteges, who may be closer to your rank and more likely to communicate with you. You may build a relationship with them, and learn a thing or two about their interaction with their mentor.

Try technology

For their new publications... nowadays most online journals allow leaving comments online. You may try to say a nice thing or two there. If they write a blog, that's even easier. Some researchers maintain a LinkedIn page or a Twitter account, try connect with them and follow them. Hope for the best.

Use other famous people as leverage

If you really want to communicate with them, also try using other organizations. For instance, you can write to some online radio station and suggest an interview topic and some guests, which of course will include your idolized researcher. You can also write to some prominent podcast hosts and give them a couple reason to invite so and so for an interview. Make good use of crowdsourcing, invite your peers and friends to support your petition.

Some heroes/heroines are better left a bit mysterious

This is sad but occasionally painfully true. Some famous people are better left not known at personal level. They could be immensely arrogant, they may not have a nanogram of social skill, they may be a jerk... Unless I have reconstructed a pretty concrete and reliable image about the researcher from different sources, I would probably want to keep them as what they are in my mind, and as an inspiration for my work.