I had cut back the outside sheathing to inspect the damage and it appears that only the red wire was pierced. The others look intact. In order to get power restored, I ran to HD and used a split bolt connector and wrapped it taught in the 3M rubber tape extra thick all around and then the vinyl tape. This was a temporary solution,as it is not in a junction box. I am now to seeing what my options are short of having to run a brand new 6/3 line.

Since this cable runs through a utility room in the basement, I could possibly cut the cable here and connect in an approved method and run a new wire from here to the breaker, since the damaged occurred between these 2 points. Not sure what the appropriate fix would be?

I have seen the Polaris connectors mentioned. Not sure if this is the best method or if these connectors need to be put in a junction box or not.

All options would be appreciated :).

Either remove the damaged section and run a new piece back to the panel, or insert two junction boxes with new cable between them to provide adequate working wire in the pair of new boxes to make connections.

1) What is the best method to make the splice/repair? Polaris connectors? Just Split Bolts? Other?

2) 6ga cable is thick so I assume it probably needs allot of room for a bend radius, etc for slack and termination. Especially with three wires plus the ground. What do I use for the box, especially size, to accommodate making all of these terminations? Didn't know if there was a box like a disconnect box that could be used to make the connections but without the disconnect per se. Although not sure if something that large would be needed.

You just need a junction box large enough.

The wire fill tables and bend radius data is in the code, you just need to look it up.