We'll until I see evidence that a perfect mirror exists, i'll assume we don't have access to one and say it gets absorbed rather quickly by human standards.

Drop the sphere and take a one dimensional case to simplify the geometry. Two mirrors seperated by one nanolight second (about 1 foot). Now this is a REALLY REALLY good mirror 99.9999% reflective so our test photon has a 1/million chance of getting tansmitted. Now we let the photon go between the plates. How long before it get's transmitted into an absorbing material, about 1 milisecond.

This effect is used in certain gas LASERs like the He-Ne and to add to the efficency, the spacing of the mirrors is an integer number of the LASERs peak output wavelength.

Ask R.L., I'm sure he could tell you more than he ever wanted to know...

How would a laser beam act if shot in such a sphere?

I dunno, but I bet it'd be really cool with a fog machine.