But with this safe haven being inside Russia, they are hesitant to share those stories.
Alexey Nechipurenko, 45, was maimed as Russian forces entered the southern port city of Mariupol. His foot was shot to pieces and his wife was killed before his eyes, he tells CNN.
But, as a Russian doctor tends his wounds, he insists Ukraine, not Russia, is to blame for his suffering.
“The Russians were just beginning to enter the city. Therefore, they just couldn’t actually have been on the side where we were,” he told CNN.
The basketball court shelter is in Taganrog, southern Russia, just 69 miles from Mariupol where Ukrainian soldiers and civilians held out for weeks in the Azovstal steel plant before Russia took full control of the city.
CNN was given exclusive access to the center set up to process some of the more than 2 million refugees estimated to have poured onto Russian soil since the invasion began on February 24.
Human rights groups say Ukrainians are being “filtered” before being taken to the temporary shelters in Russia and any suspected of posing a threat are not allowed through.
And those who passed Russia’s first test and made it to Taganrog are reluctant to say too much.
“Now I’m here [in Russia] so please don’t press me, said a 30-year-old man from Mariupol who asked not to be identified and only wanted to be recorded talking to CNN with his back to the camera.
“I didn’t see who killed my relatives,” he said. “As far as I’m concerned, they’re just a casualty of this conflict,” he added.
Dmitry Vaschenko, an official with Russia’s Ministry of Emergency Situations in Taganrog, said housing would be given to Ukrainians, who were also free to seek work and send their children to school.
“When hostilities end in the future, all these arrivals can make the decision to return to their homeland. Whoever wishes to remain in Russia, the Russian government takes such an obligation — they will receive a full range of social services and are protected,” he said.
When asked about the process to allow refugees into Russia, he said there were “filtration points” on the border.
“They are checking people who appear aggressively disposed towards the Russian Federation,” he said. “Filtering occurs precisely upon arrival, there are no ‘mass camps.’ They are border-crossing points, nothing more.”
Across the gymnasium sits another refugee from Mariupol — Irina, who fled with her nine-year-old son Rostislav and their cat Bolik. She said their city is in ruins but chooses not to apportion blame.
“I don’t want to get messed up in all of that. This side’s not right, and that side’s not right. Both sides are guilty. Both sides have shelled us. Both have killed us,” she said.
Different country, different story from refugees
The only safe route out of Mariupol for Irina was to Russia, but she hopes to move on to a third country.
Many Ukrainians have made it through Russia to Estonia, once a part of the Soviet Union, now independent and a member country of the European Union.
On board the Isabelle, a giant passenger ferry now offering shelter in Tallinn, refugees talk more freely, and tell CNN how they made it through and out from Russia and its system of filtration camps.
Daniil, 22, who feared being conscripted to fight against Ukraine, said he pretended he wanted to make Russia his permanent home. He said he was stripped and had his tattoos inspected.
“They checked if I was involved in any way with the Ukrainian army and if I know anyone who is serving there,” said Daniil, who also used to live in Mariupol.
“They asked if I know when Vladimir Putin’s birthday is, as ‘He is your president now,’ they said.
“I told them I did not know and they confronted me about my lack of knowledge,” Daniil continued. “They said ‘You must know it.’ I had to tell them that I did not have the opportunity to find that out yet but reassured them that I will learn it. So, they let me through.”
Stanislav and Vitalina, a young married couple, had thought their small city of Rubizhne might escape the worst of the war as they believed it was not strategically important. But as the battle for nearby Severodonetsk intensified in early May, the fighting came to their door and the town was occupied.
“There was no possible option to get to the Ukrainian side from our town. No one would dare to cross through an active battlefield,” Stanislav said.
Vitalina added, “For us the main thing was to save ourselves and our family, that is why, unfortunately, we had to go through Russia.”
The couple decided to pretend they were on their way to visit relatives.
“We had to answer various questions about our political views, if we support our army and why we are not supporting our army,” Vitalina said.
“During the questioning they took my phone and had it in their hands throughout the whole time, they went through my bank accounts, personal photos, and messages. Those are my personal things, and they went through all of it.”
With tears in her eyes, she talks of having to hide her hatred towards Russia while there. Now in Estonia, she reveals her true feelings.
“They tortured our people there. They kicked people out of their homes or simply did not even let us have any water. They told us that was the payback for eight years of their suffering and now it is our turn to suffer,” she said, referring to the long-running and deadly fight in the east of Ukraine between Russian-backed separatists and Ukrainian forces.
It wasn’t just the people who suffered, Vitalina said.
“Russians determined that dogs would bark at them and give away their positions, so they decided to kill all the pets,” she said.
“We would tie up our dog and put a muzzle on it, but still they killed my dog … My father confronted the soldiers who killed our pet and in return they opened fire. My dad luckily managed to get behind the house in time.”
The couple’s parents are still in Russian-occupied Ukraine. Vitalina said her father has been shot and injured and her grandfather is too infirm to leave.
They want to go back to them, to return home, but there is little hope for that right now.
“My soul longs to return home, to my family. But I understand the realities,” Vitalina said. “Everything is destroyed, there is no work, no food. Everything costs five times its original price. People are not able to survive.”