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It can definitely be repaired, it would need extensive spackling, most likely with 2 part Bondo, then sended primed and painted.
this is something that should not take more than 6-8 hours in total, obviously it does require some skill.
The picture shows the door is stained wood. The wood is repairable but the door will have to be painted or the repairs will be visible.
Do you know if the estimate from Home Depot was to replace the entire door assembly or just the (door(s)?). If they estimated the cost including removal and replacement of door jambs, threshold, and door trim, then an alternative that could cost less would be to replace the door slabs only and keep the existing jambs and trim.
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