Sfdwhf

We find these verses in 16th chapter of Gita.

tān ahaṁ dviṣataḥ krūrān saṁsāreṣu narādhamān kṣipāmy ajasram
aśubhān āsurīṣv eva yoniṣu

āsurīṁ yonim āpannā mūḍhā janmani janmani mām aprāpyaiva kaunteya
tato yānty adhamāṁ gatim

English translation by Swami Prabhupada:

BG 16.19: Those who are envious and mischievous, who are the lowest
among men,I perpetually cast into the ocean of material existence,
into various demoniac species of life.
BG 16.20: Attaining repeated birth amongst the species of demoniac
life, O son of Kuntī, such persons can never approach Me. Gradually
they sink down to the most abominable type of existence.

English Translation By Swami Gambirananda

16.19 I cast for ever those hateful, cruel, evil-doers in the worlds, the vilest of human beings, verily into the demoniacla classes.
16.20 Being born among the demoniacal species in births after births, the foods, without ever reaching Me, O son of Kunti, attain
conditions lower than that.

English Translation By By Dr. S. Sankaranarayan

16.19. These hateful, cruel, basest men, I hurl incessantly into the inauspicious demoniac wombs alone in the cycle of birth-and-death.
16.20. Having come to the demoniac womb, birth after birth, and not attaining Me at all, these deluded persons, therefore, pass to the
lowest state, O son of Kunti !

English Translation By Swami Sivananda

16.19 Those cruel haters, worst among men in the world, I hurl those evil-doers into the wombs of demons only.
16.20 Entering into demoniacal wombs and deluded, birth after birth, not attaining Me, they thus fall, O Arjuna, into a condition still lower than that.

English Translation by Shri Purohit Swami

16.19 Those who thus hate Me, who are cruel, the dregs of mankind, I condemn them to a continuous, miserable and godless rebirth.
16.20 So reborn, they spend life after life, enveloped in delusion. And they never reach Me, O Prince, but degenerate into still lower forms of life.

Krishna seems to suggest eternal samsara and all translated those verses that way.

Do these verses of Gita support eternal cycle of birth and death for some living beings and they have no way out?

Samsara = bhumi + svarga + naraka

Samsara is eternal. Creation and Destruction have always happened and will always happen. 'yatha purvam akalpayat' - RigVeda 10.190.3

Krishna says :

I forever punish evil men

You interpreted :

I punish evil men forever

Krishna says :

Evil men never reach me

You interpreted :

Evil men never reach me

Good or Evil are characteristics of an action, not of a person.

If a person does a good action, we 'temporarily' label him as good person.

If a person does a evil action, we 'temporarily' label him as evil person.

You're wrongly assuming that once a person is labeled Good or Evil, they remain that way forever. That's not necessarily true.
Because after they finish facing consequences of action, the person goes back to blank slate (kind of, cos vasanas of action still exist, and these powerful vasanas propel you to do same acts as before).

Bhagavan does not damn sinners to hell forever.

Bhagavan does not deign saints to heaven forever.

Both are temporary. After exhaustion of the rewards or punishments, the Jiva has to return to earth. 'kshine punye martya lokam vishanti' - Bhagavad Gita 9.21

Since Samsara is eternal, there will always be saints, sinners and normal people.
As long as sinners exists, Krishna will do his duty of punishing them. Duration of punishment depends on type of sin.

The Jiva, out of his own free-will, can choose to stay in samsara, or get out. But only the will is free, the consequences are not. That is in Bhagavan's hands.

Where he stays within samsara, what happens to him during his stay, and how long he stays, is a combination of his free-will to act, and Bhagavan's free-will to react.

The whole point of Bhagavan using such harsh words in Gita is to drive home this point :

A person, as long as he does evil acts, cannot reach Me.

Sri Krishna says :

9.17 Of this world I am the father, mother, ordainer, (and the), grand-father; I am the knowable, the sancitifier, the syllable Om as also Rk, Sama and Yajus.

9.18 (I am) the fruit of actions, the nourisher, the Lord, witness, abode, refuge, friend, origin, end, foundation, store and the imperishable seed.

He, Who is the Mother, Father and Friend of ALL is certainly not that cruel to retain some people in the world of suffering eternally. And, He is not only the ordainer but also controls every action staying in the heart of ALL. (18.61).

It is obvious that to Him, none is detestable and HENCE EVIL also:

9.29 I am impartial towards all beings; to Me there is none detastable or none dear. But those who worship Me with devotion, they exist in Me, and I too exist in them.