What do you mean by "half the size of Earth" when you describe that as a desired goal?
The Planet Earth has a radius of 6,371 Kilometers and a diameter of 12,742 Kilometers. A planet with half the diameter of the Earth, or 6,371 kilometers, would have one eighth the volume. If that planet had the same average density as Earth it would have one eighth (0.125) the mass of Earth.
For a planet to have one half the mass of Earth and the same average density as the Earth, it would have to have half the volume of the Earth. Thus it would need to have approximately 0.7937 times the diameter of Earth, about 10,113.3254 Kilometers, to have a volume of about 0.499999006 that of Earth.
Compare those figures with the minimum masses for a planet to keep and/or to produce a breathable oxygen rich atmosphere which are given below.
Long, long ago, back in 1964, a book was published with a scientific discussion of what is necessary for a planet (or other world) to be habitable for humans.
Habitable Planets for Man, Stephen H. Dole, 1964, 2007. I don't know if the 2007 edition was updated with more recent scientific information.
https://www.rand.org/content/dam/rand/pubs/commercial_books/2007/RAND_CB179-1.pdf[1]
There have been many more recent discussions of the habitability of other worlds using more recent and advanced science. But as far as I know most or all of those discussions are about habitability for life in general, not habitabilty for the more specific case of humans and life forms with similar requirements. On Earth, for example, many or maybe even most lifeforms flourish where humans would swiftly die.
On pages 53 to 58, Dole discusses how massive a world would have to be to retain a dense enough atmosphere of oxygen. On page 54 Dole concludes that a planet would have to have an escape velocity of 6.25 kilometers per second to retain an oxygen atmosphere for geological time periods. That corresponds to a planet with:
a mass of 0.195 Earth mass, a radius of 0.63 Earth, and a surface gravity of 0.49 g.
A radius of 0.63 Earth is a radius of 4,013.73 kilometers or a diameter of 8,027.46 kilometers.
Dole believed that a planet of that size could retain an oxygen rich atmosphere, but could not produce one. If Dole was correct, a planet of that size could only have an oxygen rich atmosphere if it was terraformed to have such an atmosphere by a highly advanced society.
Dole made two different calculations of the minimum mass that might be necessary for a world to not only retain an oxygen rich atmosphere but also to produce one. One was a mass of 0.25 Earth mass, and the other was a mass of 0.57 Earth mass. Dole considered those masses to be inaccurate, and settled on a mass of 0.4 Earth mass as the minimum mass required to produce an oxygen rich atmosphere.
This corresponds to a planet having a radius of 0.78 Earth radius and a surface gravity of 0.68 g.
A radius of 0.78 Earth radius is a radius of 4,969.38 kilometers and a diameter of 9,938.76 kilometers.
Mars has a mass of 0.107 Earth mass, a radius of 3,389.5 kilometers, and a diameter of 6,779 kilometers, so any world massive enough to retain and/or to produce an oxygen rich atmosphere should be significantly more massive and large than Mars.
Until and unless a science fiction writer finds a later and better set of calculations than Dole's they should not write about a planet with an oxygen rich atmosphere breathable for being similar to humans unless it has mass of at least 0.195 Earth and a diameter of at least 8,027.46 kilometers. And if they don't want the planet to have an artificial oxygen rich atmosphere created by a highly advanced civilization but have a naturally formed oxygen rich atmosphere instead, they should make their world have a mass of at least 0.4 Earth mass and a diameter of at least 9,938.76 kilometers.
And of course either minimum mass would be significantly greater than the mass of Mars, 1.822 or 3.738 times the mass of Mars. And also significantly less than the mass of Venus, 0.239 or 0.490 that of Venus. L Dutch - Reinstate Monica used a planet with the mass of Venus, 0.815 that of Earth, to calculate the Hill sphere of the planet in his answer.
I note that the size of your planet's Hill Sphere, will depend on the planet's mass, the distance to its star, and the mass of the star. I also note that a moon can have a stable orbit only within about 0.5 to 0.666 of the outer edge of the Hill Sphere.
https://en.wikipedia.org/wiki/Hill_sphere#True_region_of_stability[2]
I am not certain that two moons could have stable orbits around the least massive possible habitable planet at the distances indicated in L Dutch - Reinstate Monica's answer.
The Hill Sphere of Earth extends to about 1,500,000 kilometers, so the zone where moons can have stable orbits should extend to about 500,000 to 750,000 kilometers.
https://en.wikipedia.org/wiki/Hill_sphere#Formula_and_examples[3]
The example in L Dutch - Reinstate Monica's answer has two moons orbiting at about 100,000 and 400,000 kilometers, and both would be within the stable orbital zone of Earth. However, the question asks for a planet as small as possible, and L Dutch - Reinstate Monica mentioned a planet slightly larger than Mars earlier in his answer.
A planet significantly smaller than Earth would have a smaller Hill Sphere than Earth, and the moons would have to orbit closer. But if the planet is less massive, moons of a specific mass will have larger Hill spheres, perhaps interfering with one another.
Perhaps L Dutch - Reinstate Monica should recalculate his orbits for a planet massive enough to have an oxygen rich atmosphere orbiting a more massive and brighter star than the Sun at a greater distance than Earth orbits the Sun, to find a stable orbital configuration.
