The most common and most likely answer to this crossword clue is the 7 letter word CHINESE
I've seen this crossword clue in Independent
C
H
I
N
E
S
E
More clues leading to the result CHINESE
- Asian revolutionary accommodating fashionable French art
- More than a billion people found in machine search
- Backbone shown by South-East Asian
- Eastern tea joints, by the sound of it?
- Language used by machines extensively
- Unsettled, since he is from a far country
- Awfully nice, she is from an Asian country
- Extraordinarily nice, she is from a far country
- Famously moreish grub? Feasted on and off, after having part by mouth
- Language stored in machines electronically
Buddy explains!

The straight part
We know that "Eastern" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case eastern = chinese.The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.