The first manned spacecraft that humans launched was Vostok 1 in 1961, which orbited the earth for 108 minutes (roughly two hours). Using this as the baseline of a 'spaceship', a human could likely have survived in space as early as the 4th century BC.
A person inhales and exhales roughly 15 cubic feet of air per hour. To survive two hours, a person would need 30 cubic feet of air. With a typical adult male human body occupying a volume of ~3 cubic feet, a sphere of four foot diameter would easily hold a person and enough air to survive two hours in space.
Diving bells were used as early as the 4th century BC, which means that technology of the time could produce a shell that was air tight.
Since the spacecraft wouldn't need to resist the water pressure a diving bell would need to resist, it's possible that some type of airtight(ish) bladder / balloon could be large enough to hold both a person and two hours of air at a much earlier time.
Even if the bell / bladder wasn't completely airtight, coating the bell / bladder with pitch would have provided enough of a seal to prevent too much air loss.
After two hours CO2 buildup in a craft of this volume would start to pose a danger, but the occupant could certainly survive such a flight. An arbitrarily larger bell / bladder could extend the time of this flight almost indefinitely.
Since vacuum is an excellent insulator, an additional heat source wouldn't be necessary. Furs or other heavy clothing would be sufficient to keep the occupant of the craft warm.
As the craft becomes larger and the outside surface area increases, the body heat of the occupant could not extend the flight indefinitely. Without an additional heat source, the body heat of the occupant could not extend the flight indefinitely.
Still, with careful heat management, good insulation, and perhaps preheated objects to supplement the occupant's body heat, the occupant could perhaps survive a full day before the craft radiated too much heat and the occupant froze to death.
Who cares? Worrying about an increased risk of cancer in old age is a thoroughly modern concept.
This is a tough one with traditional power sources, but we don't have the same limitation here. A limitless power source would allow the craft to re-enter the atmosphere at an arbitrary speed, so re-entry heating issues don't need to be considered.
Additionally, if care were used the heat problem above could be mitigated (using atmospheric friction to warm the craft) making the volume of the craft (and air contained within) the primary limiting factor of flight time.
For a two hour round trip, a person would be able to survive with nothing more than air and warmth. Additional comforts like light, food / water, or hygiene would be secondary and could be addressed in one way or another. The occupant wouldn't have to be happy or comfortable, but people can survive a significant amount of discomfort.
This certainly wouldn't be the safest or most comfortable way to travel into space, but historically the value of life hasn't been particularly high. An ancient general / civilization would probably think little of sending dozens or hundreds of men to their deaths in a such a manner, and in all likelihood many would survive their trips into space.
Time to and from orbit might be a concern as well, as such craft wouldn't be able to travel at high rates of speed in the lower atmosphere, but it probably wouldn't change this considerably.
The actual mechanics of this craft would, of course, be dictated by the nature of the magical propulsion system.