On a question like this it is best to start with definitions of terms as opinions won't reliably get us very far.
"Epical," when applied to prose, describes a work of prose that follows a (usually long) sequence of events that involve heroic actions and generally has some sort of triumphant end. To illustrate, Homer's Iliad is an epic poem and a model for determining "epical" as applied to prose.
"Lyrical," when applied to prose, describes a work of prose that is melodic in quality (songlike) and the reveals thoughts and feelings. While prose is not poetry, lyrical prose is said to be "poetical" in quality.
Firstly, then, according to the definitions, this story is not epical because it does not cover a dramatically long period of time. It covers an incident that only involves many days. It also is not replete with acts of heroism, unless trying to get information from an unwilling bureaucracy about an insignificant non-citizen is heroic (and it may be ...).
Secondly, then, according to the definitions, this story is lyrical because the language is poetical and it is replete with the narrator's thoughts and feelings.
- You long to hear nothing but a deep satisfying silence when you sound a marriage.
- white ducks on the dam, the lucerne field brilliant as window-dresser's grass ....
Thoughts and Feelings
- The farm hasn't managed that for us, of course, but it has done other things, unexpected, illogical.
- for those moments they looked exactly alike, though it sounds impossible.
- I got up awkwardly as she watched me--how is it I always feel a fool when I desert her bed?
Thus from the applications of the definitions of the terms and the analysis of the text in light of those definitions, it is fair to say that "Six Feet of the Country" is lyrical prose and not epical prose.