Μετά διπλώνω τη σακούλα και τη βάζω πίσω στην αποθήκη.

Because Greek often leaves out subject pronouns when they are clear from the verb ending.

• διπλώνω = I fold
• βάζω = I put

So the subject I is already built into the verbs. You could add εγώ if you wanted emphasis, but it is not necessary:

• Εγώ διπλώνω... = I fold... / As for me, I fold...

Here Μετά means then or afterwards. It is being used as an adverb to show sequence.

Putting it first is very natural because it sets the time/order for the whole sentence:

• Μετά διπλώνω... = Then I fold...

Greek word order is fairly flexible, so you may also see διπλώνω μετά... in some contexts, but sentence-initial Μετά is very common when narrating steps.

Also, this is different from μετά used as a preposition, as in:

• μετά το μάθημα = after the lesson

Greek often uses the present tense for:

• habitual actions
• instructions
• step-by-step descriptions
• vivid narration

So Μετά διπλώνω τη σακούλα και τη βάζω... can sound like:

• Then I fold the bag and put it back...
• Next, I fold the bag and put it back...

This is very natural if someone is describing a routine.

Also, both verbs are 1st person singular:

• διπλώνω = I fold
• βάζω = I put

Σακούλα is the direct object of διπλώνω, so it is in the accusative case.

The noun is feminine singular:

• nominative: η σακούλα = the bag
• accusative: τη(ν) σακούλα = the bag

Since the speaker is doing something to the bag, Greek uses the accusative.

The first τη is the article:

• τη σακούλα = the bag

The second τη is a weak object pronoun:

• τη βάζω = I put it

So the sentence literally works like this:

• Then I fold the bag and put it back in the storeroom.

Greek very naturally repeats the object with a pronoun when the noun has already been mentioned.

In Modern Greek, the feminine singular accusative article/pronoun can appear as τη(ν). The final -ν is often dropped before many consonants.

So both of these underlying forms are related:

• την σακούλα
• την βάζω

In normal modern usage, they are usually written here as:

• τη σακούλα
• τη βάζω

The final -ν is more likely to stay before:

• vowels
• κ, π, τ, ξ, ψ
• consonant clusters like μπ, ντ, γκ, τσ, τζ

That is why you also get:

• στην αποθήκη and not στη αποθήκη, because αποθήκη begins with a vowel.

Because weak object pronouns in Modern Greek normally come before a finite verb.

So:

• τη βάζω = I put it
• τον βλέπω = I see him
• τους ξέρω = I know them

This is one of the big differences from English, where object pronouns come after the verb:

• I put it
• not I it put

An important exception: with affirmative imperatives, the pronoun usually goes after the verb:

• Βάλ’ τη πίσω. = Put it back.

Πίσω basically means back or behind, but here it means back, not necessarily behind.

In this sentence:

• τη βάζω πίσω στην αποθήκη

the meaning is:

• I put it back in the storeroom
• I return it to the storeroom

So πίσω adds the idea of returning something to its previous place.

Compare:

• είναι πίσω από το σπίτι = it is behind the house
• το βάζω πίσω = I put it back

Στην is the contracted form of σε την.

• σε = in / at / to
• την = the (feminine accusative)

So:

• σε την αποθήκη → στην αποθήκη

This contraction is standard and very common in Greek.

Also, αποθήκη is feminine, so the article has to match it:

• η αποθήκη = the storeroom / storage room
• στην αποθήκη = in/to the storeroom

Modern Greek σε covers several meanings that English often separates:

• in
• at
• to

The exact meaning depends on the verb and the context.

So:

• είμαι στην αποθήκη = I am in the storeroom
• πάω στην αποθήκη = I go to the storeroom
• τη βάζω στην αποθήκη = I put it in the storeroom

In your sentence, because the verb is βάζω (put), στην αποθήκη expresses the destination/place where the bag ends up.

Greek word order is more flexible than English, but not completely free.

This sentence has a very natural neutral order:

• Μετά διπλώνω τη σακούλα και τη βάζω πίσω στην αποθήκη.

You may see variations for emphasis, for example:

• Μετά τη σακούλα τη διπλώνω...
  This gives extra emphasis to the bag.

But some things are fairly stable:

• the weak pronoun usually comes before the finite verb: τη βάζω
• Μετά often comes early when marking sequence
• πίσω naturally stays near βάζω

So yes, word order can change, but the original sentence is a very normal, unmarked way to say it.