How, if possible, can a species be saved from extinction if the only surviving population are males that cannot reproduce with each other?
The following answer only deals with theorical feasability
Can a species populated solely by males be saved from extinction?
Short answer : No, not with certainty
Nuanced : yes, in some species, it could be possible
It varies in function of how you determinate the sex of your species. Sex determination is a wide topic, see the appropriate wikipedia page
Some reptilian species are not sexually determined by chromosomes but by temperature. An example is the painted turtle, of which thermal sex determination has been studied by Carrie Lynne Morjan in an article titled Temperature-dependent sex determination and the evolutionary potential for sex ratio in the painted turtle, Chrysemys picta
Thermal sex determination allows you to get 2 different sexes from the "same" genome, as temperature will be the driver of the hormonal differentiation.
Cross-species cloning: influence of cytoplasmic factors on development, by Sun and Zhu states that cross species cloning has been successfully made with fishes and mammals.
Nevertheless, cross-species NT has succeeded in cloning some endangered mammals, such as the gaur
In fish, a type of relatively primitive vertebrate, cross-species NT [nuclear transfer] could be achieved in quite a few genetically distant species
It means we are able to use as a mother and egg-layer a different, albeit related, species.
We can, with a massive grain of salt, hypothesize that we could, in a specie solely constituted of sperm producing male, use the oocyte of another specie to clone it, then use the thermal differentiation to make a female out of our clone.
However, I am not aware of any information about how much genome and spermatozoids genetic payload is modified by the sex differentiation. Therefore, it is possible that sex-determined genes cannot produce a sexually naive clone. For more information, and if you want to research more about it, I recommend you to see the current state-of the art in de-differenciation technics and DNA-demethylation for cloning.
I hope I provided a sufficient answer for you.