Пожарный повторил, что при эвакуации надо идти спокойно и не возвращаться за вещами.

Yes. Пожарный originally means fire-related as an adjective, but it is also very commonly used as a noun meaning firefighter.

So in this sentence, пожарный = the firefighter.

A learner may also hear пожарник, but пожарный is the more standard and neutral word.

Повторил is:

• past tense
• masculine singular
• perfective

It agrees with пожарный, which is masculine singular.

The perfective verb повторить means to repeat once / to say again as a complete action. So пожарный повторил means the firefighter repeated / restated.

If it were повторял, that would be imperfective and would suggest something more like was repeating or used to repeat.

In Russian, что often introduces a subordinate clause, just like that in English.

So the sentence structure is:

• Пожарный повторил = the firefighter repeated
• что ... = that ...

Russian normally puts a comma before this kind of subordinate clause:

• Пожарный повторил, что ...

При эвакуации means something like:

• during an evacuation
• in the event of evacuation
• when evacuating

The preposition при often means during, under, or in the context of.

So this is not about movement to an evacuation. It means the rule applies when an evacuation is happening.

Because при requires the prepositional case here.

The noun is:

• nominative: эвакуация
• prepositional: эвакуации

So:

• при эвакуации = during an evacuation

This is a very common pattern:

• при пожаре = in case of fire / during a fire
• при встрече = upon meeting
• при необходимости = if necessary

Надо is an impersonal word meaning:

• it is necessary
• one should
• you must / should

Russian often leaves out the subject in general instructions. English often uses you, but Russian does not need to.

So:

• надо идти спокойно
  literally = it is necessary to go calmly
• natural English = you should walk calmly

This makes the instruction sound general and applicable to everyone.

This is a classic Russian motion-verb contrast.

• идти = to go on foot in one direction, in one specific situation
• ходить = to go on foot habitually, repeatedly, or in various directions

An evacuation is a specific movement happening right now or in one concrete situation, so идти is the natural choice.

So идти спокойно means to walk/go calmly in that evacuation situation.

Because it modifies the verb идти.

• спокойный = adjective, calm
• спокойно = adverb, calmly

Here the meaning is go calmly, so Russian needs the adverb:

• идти спокойно

If you used спокойный, it would need to describe a noun, not a verb.

With instructions, rules, and general prohibitions, Russian often uses the imperfective infinitive.

So:

• не возвращаться = not to go back / not to return
• this sounds like a general instruction or prohibition

That fits the context of evacuation rules very well.

A perfective form like не вернуться would sound less natural here, because it focuses more on a single completed return rather than the general action of going back.

In this verb, -ся is part of the verb meaning to return / to go back.

Compare:

• возвращать = to return something, to give something back
• возвращаться = to return oneself, to go back

So here:

• не возвращаться = not to go back

For learners, it is best to memorize возвращаться as a whole verb meaning to return.

За вещами uses the preposition за with the instrumental case.

Here за means for / to get / to fetch after a verb of motion:

• идти за врачом can mean go get the doctor
• вернуться за сумкой = go back for the bag
• возвращаться за вещами = go back for things / belongings

The noun is:

• nominative plural: вещи
• instrumental plural: вещами

So вещами is instrumental plural.

Literally, вещи means things, but in this context it usually means belongings, personal items, or possessions.

So не возвращаться за вещами means:

• don’t go back for your belongings
• don’t return to get your things

In an evacuation context, that is a very natural use of вещи.

Because one надо can govern both infinitives.

The structure is:

• надо идти спокойно
• и не возвращаться за вещами

Together, that means:

• you should go calmly and not return for your belongings

Russian does not need to repeat надо before the second verb, although repeating it would still be understandable. Using it once is smoother and more natural.