"This Old Starship" is a running joke performed by both Adam and Ben in which they affect Boston (BAH-STON) accents similar to those of the hosts of the long-running PBS television show "This Old House," most notably the exaggerated accent of Norm Abram, master carpenter and flannel model.
Ben and Adam apply architecture and house building terminology as used by the TOH hosts to sets and areas of the Enterprise while trying to maintain cartoonishly Abram-esque Boston accents.
Typically, the bit begins with an introduction similar to "Now if you're going to eject the warp core,..". Like most running impressions on the show, it is usually accompanied by background music.
Penny nails are often discussed.
Some unaware viewers have assumed that Ben is trying to do a Peter Griffin accent. They are wrong.