The Fitbit Aria scales (older model FB201B) seem to have a design flaw which causes the plastic part in the two front feet to stress fail after a relatively small number of uses (typically just after the warranty expires). Once broken, the scale doesn't detect a zero load causing the display to loop between 'step off' and 'sensing' leading to battery drain. There are no replacement parts, rendering a $120 scale useless for the sake of a 0.10c piece of plastic. It is relatively easy to open (see comments for video) and fit this part which corrects that flaw.

UPDATE:
Lasted for 3 years. Just printed another one.

To open the case, remove the two screws in the battery compartment and pry the back off along the outer edge of the scale.