Η μεταχειρισμένη αντίκα είναι παλιά.

They do not mean the same thing.

• μεταχειρισμένη means used, second-hand, or pre-owned
• παλιά means old

So the sentence is describing the antique in two different ways:

• it is used / second-hand
• it is old

This can sound a bit redundant in English, because an antique is normally old already, but grammatically the Greek sentence is fine.

Η is the feminine singular definite article, meaning the.

Greek articles change for gender, number, and case. Here:

• η = the for a feminine singular noun in the nominative case

It matches αντίκα, which is treated as a feminine singular noun here.

Because it has to agree with αντίκα.

In Greek, adjectives agree with the noun they describe in:

• gender
• number
• case

Since αντίκα is feminine singular nominative, μεταχειρισμένη must also be feminine singular nominative.

That is why you see the feminine form μεταχειρισμένη rather than a masculine or neuter form.

In this sentence, yes, αντίκα is treated as a feminine noun. The ending -α often goes with feminine nouns, but not always in every case across Greek.

Also, αντίκα is a loanword, and words like this are sometimes a little special in how they behave. Even so, in everyday usage it is commonly treated as feminine, so you get:

• η αντίκα
• η παλιά αντίκα
• η μεταχειρισμένη αντίκα

Because here παλιά is a predicate adjective, not an adjective placed directly before the noun.

Compare:

• η παλιά αντίκα = the old antique
• η αντίκα είναι παλιά = the antique is old

In your sentence:

• Η μεταχειρισμένη αντίκα = the used antique
• είναι παλιά = is old

So παλιά comes after είναι because it is part of the predicate.

For the same reason as μεταχειρισμένη: it describes αντίκα, so it must agree with it.

Even though παλιά comes after είναι, it still refers back to η μεταχειρισμένη αντίκα. Because that subject is feminine singular, the adjective is also feminine singular:

• masculine: παλιός
• feminine: παλιά
• neuter: παλιό

Because Greek verbs do not agree with gender in this kind of sentence. They agree mainly with person and number.

So:

• είμαι = I am
• είσαι = you are
• είναι = he/she/it is

The subject may be masculine, feminine, or neuter, and είναι stays the same:

• Ο άντρας είναι ψηλός.
• Η γυναίκα είναι ψηλή.
• Το παιδί είναι ψηλό.

The adjective changes, but the verb form είναι does not.

It functions as an adjective, but historically it is a participle form.

Greek very often uses participle-based forms as normal adjectives. So μεταχειρισμένη behaves just like an adjective here and means used / second-hand.

For a learner, the most important thing is to recognize that in this sentence it acts as an ordinary descriptive word and agrees with the noun like an adjective.

Yes, Greek word order is more flexible than English, although not every version sounds equally natural in every context.

The basic sentence:

• Η μεταχειρισμένη αντίκα είναι παλιά.

Possible variations can shift emphasis:

• Η αντίκα είναι παλιά.
• Παλιά είναι η μεταχειρισμένη αντίκα.
  This is possible, but it sounds more marked or contrastive.

For a learner, the original order is a very natural and clear pattern:

• article + adjective + noun + είναι + adjective

The written accent mark shows the stressed syllable.

In this sentence:

• Η — no stress of its own here
• μεταχειρισμένη — stress on -μέ-
• αντίκα — stress on -τί-
• είναι — stress on εί-
• παλιά — stress on -λιά

A rough pronunciation guide:

• η me-ta-hi-ri-SME-ni an-DEE-ka EE-ne pe-LYA

That is only approximate, but the accent marks are your best guide to stress.

The subject phrase Η μεταχειρισμένη αντίκα is in the nominative case, because it is the subject of the sentence.

Also, the predicate adjective παλιά is nominative too, because after είναι Greek normally uses the nominative for this kind of complement.

So the sentence structure is:

• subject in nominative
• είναι
• predicate adjective in nominative

It is grammatically correct, but it may sound a little unusual semantically because αντίκα already suggests something old.

So in real life, a speaker might be more likely to say something like:

• Η αντίκα είναι παλιά.
• Η αντίκα είναι αυθεντική.
• Η μεταχειρισμένη καρέκλα είναι παλιά.

Still, your sentence is useful for learning because it clearly shows:

• article-noun agreement
• adjective agreement
• the verb είναι
• a predicate adjective after the verb