In the highly unlikely event that it comes up in the interview, you simply say "<What I did and why this demonstrates my ability to increase your company's revenue or reduce its costs> The firm subsequently went out of business."
Many, many people in our industry have resume line items from organizations which are no longer going concerns. One would hope your recruiter has adjusted to this reality. If not, I might file that under "Dodged a bullet."
If you're concerned, I might add a note on your resume that this startup formally ceased operations on X date.
If those recruiters do have a problem with it, they'll ask you about it, they won't just assume you are lying and throw your resume in the garbage. I'm sure you can dig up an old W-2 from your taxes or something (in the US) to prove you did work there.
whatever.com (1 year)
This one failed, this is what I learned from that.
1) something
2) something else