Meaning of “It was a line”

I assume Joey meant that this was a pick-up line: something you say to someone to attract them - something that is not necessarily true or meaningful in any way. IOW, it is bait.

To pick someone up is to get them to go out with you, have sex with you, etc. In British English the more common expression is probably chat someone up.

The term line is often used to describe a statement that is insincere or less than true. One of the many [Oxford Dictionaries Online] definitions for the term is

informal A false or exaggerated account or story: he feeds me a line about this operation

Similarly, [Collins] defines it as

(slang) something said for effect, esp to solicit for money, sex, etc.: he gave me his usual line

And [Macmillan]

[SINGULAR] a remark, excuse, or explanation that is not sincere or true Don't give me that old line.

As @Drew notes in his answer, the term is often part of the phrase pick-up line and is used as an introduction to a romantic play.

While I have no authority for this observation, the term may have gotten this meaning from association with its use to describe a line that an actor speaks -- a statement that is not the honest thought of the speaker, but said for effect.

To answer your first question, yes, the a is an indefinite article.