Doomed? Probably, but don't be surprised if life finds a way.
Minimum viable population (MVP) is a lower bound on the population of a species, such that it can survive in the wild.
An MVP of 500 to 1,000 has often been given as an average for terrestrial vertebrates when inbreeding or genetic variability is ignored. When inbreeding effects are included, estimates of MVP for many species are in the thousands. Based on a meta-analysis of reported values in the literature for many species, Traill et al. reported a median MVP of 4,169 individuals.
Having less than 500 individuals would put almost any species in dire straits. Numbers I generally hear thrown around for humans is anywhere from 2k to 10k, but also as low as overcoming a Population bottleneck of just 70 individuals.
A 2005 study from Rutgers University theorized that the pre-1492 native population of the Americas are the descendants of only 70 individuals who crossed the land bridge between Asia and North America.
The question is, just how tiny and (more importantly how) heavily distributed the population has become and whether or not it has fallen below an Effective population size.
The harmonic mean tends to be dominated by the smallest bottleneck that the population goes through.
So for humans, research has shown that the MVP can drop as low as less than 100, but that was a localized population. I'd think that if you instantaneously distributed that same population across all 50 States, you'd have little hope of recovering from that (which IIRC, that's basically the situation we are to infer from the plot of Oblivion).
Now, as to the inhabitability of parts of the Earth, that might actually make people condense into a smaller area and thus help preserve a genetically viable population. See, Functional extinction (the state of which humanity is in IMO, in the movie).
Functional extinction is the extinction of a species or other taxon such that: [... #3] the population is no longer viable. There are no individuals able to reproduce, or the small population of breeding individuals will not be able to sustain itself due to inbreeding depression and genetic drift, which leads to a loss of fitness.
It's definitely not looking good for humanity, but extinction is far from a certainty. It's all just theories and playing with math, the only thing we can say with a reasonable amount of certainty is that (research has shown) a localized (!) population of just 70 people is, historically, a sufficient MVP.
For once, Stack Exchange was of little help researching this, as the Mathematics.SE question: Finding the critical population level has yet to of been answered.