I was afraid we would run out...

[shootingstar]

Dr. Gibbs: Julia, when I married you, do you know what one of my terrors was in getting married?
Mrs. Gibbs: Pshaw! Go on with you!
Dr. Gibbs: I was afraid we weren't going to have material for conversation more'n'ld last us a few weeks. I was afraid we'd run out and eat our meals in silence, That's a fact. You and I've been conversing for twenty years now without noticeable barren spells.
(Thornton Wilder, Our Town, Act II)

What do you take run out to mean in this context?

 

[emsr2d2]

It refers back to "material for conversation". It means he was worried that they would no longer be able to think of anything to talk about and they would eat in silence. If you "run out of" something, it means you don't have any left.

I've run out of milk.
She's run out of ideas.
They've run out of things to talk about.
We're going to run out of money quite soon.
Do we have any bread? No, we've run out.
There are only two apples in the fruit bowl. I think we might run out before I have a chance to go shopping.

It's a phrasal verb. There's a great phrasal verb list on this site. Look HERE.