Traveling the speed of light in a spaceship?

Well, technically objects that move at the speed of light do not experience time. They would not have aged, at all. But for things with mass(like humans) it is literally physically impossible to travel at the speed of light. This isn't a matter of a lack of technology, it is a fact of the universe.

They would be dead.

I calculate that moving at 17,000 MPH for a week will make you only 0.0004 seconds younger than a person not moving, not 5 minutes younger.

If you could move at the speed of light for a week you would end up 1 week younger, because time does no pass at all for things moving at the speed of light.

I would argue that time is relative... if your were you move at near the speed of light (as to date we don't know of anything able to move that fast except light (visible and non-visible) so let's say 99.99% the speed of light. your orbit would be a much wider radial arc than that of a Geostationary orbit.
A geostationary orbital  altitude very close to 35,786 km (22,236 mi), and directly above the Equator. This equates to an orbital velocity of 3.07 km/s (1.91 mi/s) or 11,000 km/hr
the speed of light The speed of light in a vacuum is defined to be exactly 299,792.458 km/s or 17987547.48 km/h - I'm not doing the math for your altitude.

it isn't actually the speed at which you move but the distance from the object you are measuring against.  

so anyone orbiting the earth at near the speed of light for a week would be a week older, but time would have passed much faster for the people of earth perhaps a month or even years - again I'm not doing the math.

As to the notion that time does not exist at the speed of light I would disagree as we can 'measure' the amount of time it takes for light to travel any given distance. Light originating on earth and travelling to the moon is 1.26 seconds older than at it's source.  If light travels through time (as we also move through time) then time is present in light, otherwise time travel both forward and backward in time would be possible at the speed of light.

Objects moving at the speed of light experience effectively zero time. So they would be exactly one week younger.

That said, there is no orbit around the Earth for which the speed is equal to the speed of light. More generally, there is no orbit around ANY object at that speed, aside from black holes.