From The Chicago Manual of Style (17th ed.), 7.5:
Most nouns form their plural by adding s or—if they end in ch, j, s, sh, x, or z—by adding es.
And from Chicago, 7.9:
Names of persons and other capitalized nouns normally form the plural by adding s or es. Exceptions, including the last example, are generally listed in Webster’s . . .
Tom, Dick, and Harry; pl. Toms, Dicks, and Harrys
the Jones family, pl. the Joneses
the Martinez family, pl. the Martinezes
the Bruno family, pl. the Brunos
Sunday, pl. Sundays
Germany, pl. Germanys
Pakistani, pl. Pakistanis
Romany, pl. Romanies
Strictly according to this guideline, and assuming that there is no entry for Barleyfoot in Webster's, the plural should be Barleyfoots.
However, the family itself is free to refer to itself in whatever manner it wishes. If a family member says, "My name is Tom Barleyfoot, but if you want to refer to my family as a whole, please call us the Dinklemeisters," then so be it.
Even if it makes no sense at all, it's the family that sets their own guidelines for their own names. The point of a proper name, if you're the person who "owns" it, is that you can do whatever you want with it. (Although the more unusual the thing you do with it, the less likely that anybody else will know to use it as you wish them to.)
So, perhaps the person speaking on the podcast actually knows that this family prefers to have people refer to them as the Barleyfeet rather than the Barleyfoots.
Or, perhaps they just made it up because it sounded right to them.
Update: A comment asked about the plural of proper names that take the form of irregular plural nouns. For instance: Fish or Wolf.
As proper names, common sense would seem to dictate that they not be treated in the same way as common nouns. However, that's a subjective take on my part. This specific circumstance isn't covered anywhere specifically. So, I guess it falls to individual preference, with the specific people in question having more of a say in it than anyone.
Note, however, that if you add "family" you leave the family name singular. This is shown in section 7.2, as quoted above, where these are equivalent: the Jones family and the Joneses; the Martinez family and the Martinezes; and the Bruno family and the Brunos.
In the same way, even if you're not sure how you should pluralize Fish or Wolf, you can still reword around it: The Fish family and The Wolf family.