How can I protect against or remove Doom once it's cast?

This is most likely the first boss in the Sunken Temple of Qarn, which periodically inflicts Doom on the entire party. This specific Doom effect can only be removed by stepping on one of three large tiles on the floor, marked by a glowing red effect.

Other types of Doom effects exist, usually the debuff's tooltip will tell you how it can be removed.

As it was random party, couldn't really ask them how they were protected or removed the status.

Of course you can ask! Most players I've met in the Duty Finder have been nothing but helpful towards newbies.

There are a total of 4 different types of Doom in FFXIV:

• Sometimes you need to step on a glowing floor tile
  • Examples: Sunken Temple of Qarn first boss, World of Darkness first boss
• Sometimes it needs to be Esuna'd off by a healer
  • Example: First boss of Dun Scaith
• Sometimes the target needs to be healed to full health
  • Example: Seat of Sacrifice
• Sometimes it's a punishment for messing up a major mechanic in a Savage raid with no way to remove it
  • I've seen this show up in one of the Eden Savage raids, although I don't recall which one

Doom (status) on the Final Fantasy Wiki - Final Fantasy XIV section:

Emphasis mine:

Doom is a status that knocks out the victim within a certain amount of time receiving it. The amount of time before it triggers depends on the user. Some variants of Doom can be removed by Esuna, some require the victim to be healed to full HP to remove, and some can't be removed at all.

There seem to be at least three different types of Doom, with some tooltips when hovered over, or however the Wiki gets the information. The icon with a blue dot appears to be the kind that goes away at full health.

Esuna in FFXIV:

Esuna is a cross-role ability available at level 20 for all healer role classes and jobs. The spell removes a single detrimental status effect from the selected target.