When downhill skiing (although I assume this would apply to cross-country skiing or snowboarding as well), when should a gouge caused by a rock or other obstacle be considered large or serious enough to take the skis in for base repair, and when can it be ignored?
The two biggest risks from gouges:
Core Shot: This is when the gouge cuts through the base material* to the core layers of the ski (usually wood or composite material). If moisture gets in between those layers (it will), it can cause the various layers of your ski to separate. This process of de-lamination will rapidly destroy your ski or snowboard and can not be repaired.
Edge Separation: If the rock catches on your edge enough to pull it out from between the core and base layers, or mangles it enough to create a gap, this can result in several issues: De-lamination (see above), affect turn performance (you have no edge / uneven edge), laceration risk (those things are sharp even when they are in proper position).
Fortunately, both of these cases can be repaired to nearly good as new if you have the tools/knowhow -- but only before de-lamination begins, so act quickly.
Beyond that, minor scrapes, dings, etc. that don't make it to the core -- even super-long and gnarly looking ones -- are fine up to your personal preference. They will effect your glide, and can affect performance, and might cause excessive snow build-up in adverse conditions, but your skis will be structurally fine.
*The polyethylene sheet that touches the snow that holds the wax that lets you glide.
If it is deep and near the edge where it can cause the metal edge to bend under use, it should be fixed. Other than that, skis (and snow boards) can handle a lot of gouging. The main problem is that it slows down the skis a little, and might affect turning if it's pretty bad. Luckily, gouges are reasonably cheap and quick to have fixed.