Low estimate: 2,525. High estimate: 25,250
A scenario like this is fine as long as the population is above a certain manageable size. Let's say that given a 1:1 male-female population, the community is required to be above the population n in order to avoid inbreeding. In this hypothetical population of 1:100 male-female, the way breeding pairs work is that every member of the population has 99 sets of half-siblings (though mostly sisters). The females just need to find a male which isn't related to them, which is simple enough to find, as any their family structure will still work the same way backwards, so all they have to do is check the prospective male's parents, grandparents, and possibly great-grandparents, and if that checks out, than it's fine. That means that this population will still need the same number of males that a 1:1 population has, but will require 100x the females. We take the '50/500' rule, which says that a population of 50 is necessary to prevent inbreeding, and 500 is needed to reduce genetic drift and apply our requisite formula - in humans it's 25 males, 25 females, so in this species its 25 males, 2,500 females; or 250 males, 25,000 females, depending on how secure you want to be. The 50/500 rule, by the way, is a very loose rule of thumb - the number itself greatly differs from species to species based on a lot of different factors but 50/500 is a good starting ballpark estimate.
Naturally avoiding inbreeding would be difficult, but for the most part it doesn't matter as long as direct incest is avoided and indirect incest (i.e. cousins) doesn't repeatedly happen, which is usually possible. (As in it doesn't matter solely from a health perspective.) If you wanted to be gimmicky about it, you could have everyone wear family crests, or something of that nature so that the other members would know whether or not they're from the same family.