Is 000… a geometric sequence?

Probably depending on the definition being used.
In general a geometric sequence to be one of the form ##a_n = a_0r^n## where ##a_0## is the initial term and ##r## is the common ratio between terms.
In some definitions of a geometric sequence (for example at the encyclopedia of mathematics) we add a further restriction dictating that ##r!=0## and ##r!=1##.
By those definitions a sequence such as ##1 0 0 0 …## would not be geometric as it has a common ratio of ##0##.
There is one more detail to consider though. In the given sequence of ##0 0 0 …## we have ##a_0 = 0##. In no definition that I have found is there any restriction on ##a_0## and with ##a_0=0## the given sequence could have any common ratio. For example if we took ##r = 1/2## the sequence would look like
##a_n = 0*(1/2)^n = 0##
which does not contradict the definition (note that the definition does not require ##r## to be unique).
So depending on the definition ##0 0 0 …## would probably be considered a geometric sequence.
Still whether ##0 0 0 …## is a geometric sequence or not is likely of little consequence as the properties and behavior of the sequence are obvious without any further classification.