When my house burned down, I got out with my two dogs and the clothes I was wearing.
If it happens again, priorities will be:
Keepsakes - photos & mementos that can't be replaced
Important Documents - wallet, passport, household inventory (last time insurance adjuster wanted to see proofs of purchase), mortgage papers,etc.
Data backup - music, photos, videos, etc. that would be difficult to replace but not impossible
And just an FYI....although nothing in the fireproof box ignited, the entire contents was turned to charcoal. Off site safe deposit box is recommended as is "Renters Insurance" (to cover your property loss and your liability when the landlord sues you for burning down his house).