It certainly does have a tune, and it's a fine ballad for a couple to sing. Jane Keefer's "Folksong Index" lists these recorded examples:

Get Up and Bar the Door
1. Gainer, Patrick. Folk Songs of the Alleghenies, Folk Heritage, LP (1963), cut#A.03
2. Gillette, Steve; and Cindy Mangsen. Light of Day, Compass Rose CRM 7, Cas (1996), cut#B.01
3. MacKay, Karen. West Virginia Woman, West Virginia Woman Rec., LP (199?), cut# 2
4. Marks, Phyllis. Folksongs and Ballads, Vol 2. Phyllis Marks, Augusta Heritage AHR 008, Cas (1991), cut#2.08
5. McCurdy, Ed. Ballad Record, Riverside RLP 12-601, LP, cut#A.04
6. Moser, Artus. North Carolina Mountain Folksongs

And I know that Ewan MacColl recorded it somewhere, too. It's Child Ballad #275, it's in the DT, and you could look it up in Bronson's Traditional Tunes of the Child Ballads, provided that your library has or can get a copy for you.

Sandy