More than 30,800 people have died from coronavirus, with the number of cases worldwide exceeding 660,000.
Shevchenko, who was a child at the time of the Chernobyl disaster in 1986 that reportedly killed 54 people and is said to have led to thousands of deaths, said he saw similarities.
"I lived a moment very similar to this. At the time, I was nine years old," the Ukraine coach told Sky Sport.
"The only thing we have to do is avoid stupid behaviour. None of us can be sure that we have no contracted the virus.
"We must not go out, we must not think only of us, but above all of those people for whom the infection would be a greater problem."
Shevchenko thanked doctors and nurses and urged people to respect guidance from their governments.
Italy, where Shevchenko played with AC Milan from 1999-2006, has seen more than 10,000 deaths due to coronavirus.
"The only solution is to respect the rules of the government – that is to stay at home and give doctors the opportunity to do their job, which is really huge," he said.
"Give them volunteer nurses, they are the heroes of our time. They must be thanked for all they are doing."
Russia invaded neighbouring Ukraine last Thursday after weeks of heightening political tensions between the two countries.
The actions of Russia have led to widespread condemnation, with financial, sporting and political sanctions imposed on the nation in an attempt to deter the attacks.
Shevchenko, who is Ukraine's all-time leading goalscorer and former head coach, has previously asked people to join him in speaking out against the attacks as he called for peace to be restored.
He has stayed in London during the conflict to help lead the humanitarian aid and raise awareness in England, but his family remain in Ukraine in solidarity with their compatriots.
For now, Shevchenko has little interest in the sport that made him famous.
"Football doesn't exist for me any more," he told Sky Sports. "I don't think about it. It's not the time for that. I'm not watching anything, any sport, anything.
"All my concentration, when I wake up, I think about how I can help my country, what I can do. I've started to call my parents, my friends, get updates on what's going on in Ukraine.
"For me, this is my field, this is my concentration now."
However, Shevchenko did note Russia's ban from FIFA and UEFA competitions, praising the decision.
"It's a great reaction from the institutions like UEFA and FIFA to make the right decision," he said.
"I don't think it's a difficult decision. When you attack a country, when you start to send in bombs and soldiers, it's not a conflict, it's a real war.
"When the war has not stopped, I think it's the right decision not to allow any Russian athletes to participate in any sporting event."
Shevchenko, 43, has been in charge of Ukraine since 2016 in his first senior coaching job.
The former forward played for the likes of Milan and Chelsea during his career, winning a Serie A title and Champions League while at San Siro.
Shevchenko hopes to one day return to Milan – as their head coach.
"Milan is always in my heart and one day, when I decide to change, I would love to train them," he said during an Instagram Live chat on Wednesday.
Milan have struggled in recent seasons, last winning the Serie A title in 2010-11.
They were seventh in the table when this campaign was suspended in March due to the coronavirus pandemic and Shevchenko hopes Milan can improve when the season resumes.
"We must try to finish the season at best at this moment. Everyone will have to give their best to have respect of the fans," he said.
"Then, we will see what will happen and what future the club will have."
An ambassador for Laureus Sport for Good, Shevchenko surprised children who had fled the war with a visit to a school in Warsaw.
The world has rallied around Ukraine following Russia's invasion, with sporting stars and celebrities using their platform to raise awareness of the situation in the country.
On Saturday, WTA world number one Swiatek organised a charity tennis match, where Shevchenko was a special guest, while Lewandowski has been vocal in his support – and will wear a Barcelona shirt featuring the UNHCR logo, the United Nations' main agency for refugees, this season.
Following his appearance in Warsaw, Shevchenko thanked Poland as a nation for taking in a flood of refugees from their European neighbours.
"I want to thank you, [the] Polish people, for being very kind to my country. For hosting us here. For providing all the needs," he said, as per Laureus Sport for Good.
"Thank you so much for your support. Also, I want to thank Iga Swiatek and Robert Lewandowski for taking a big part in initiatives helping my country to raise the funds and support against the war in Ukraine."
Shevchenko also praised compatriot Oleksandr Zinchenko for speaking out against the war, something numerous Ukrainian athletes have done.
"Like many, everyone Ukrainian, his reaction against the war, he stood up and then he spoke loud," he added of Zinchenko, who last week signed for Arsenal from Premier League champions Manchester City.
"He expressed his view, he's expressed his very hard feelings against the war for Ukraine, because he loves Ukraine, he loves people.
"And we did a lot of activities together, we've been involved in a few fundraisers. I know he's a very generous person and he did a lot for Ukrainian people."
The 44-year-old stepped up from his position as assistant to take over the Ukraine senior side in July 2016.
After missing out on qualification for the 2018 World Cup, Shevchenko guided his country to the delayed Euro 2020 finals, where they reached the last eight for the first time in their history.
However, the legendary former Milan striker announced on Sunday that he will not be extending his contract to stay on as Ukraine boss.
"Today, my contract with the Ukrainian Football Association came to the [sic] end," he posted on Instagram. "I spent five years with the national team.
"It was hard work that proved that we are capable of playing modern football. I am grateful to the president and the executive committee of the UAF for the opportunity to work with the Ukrainian National team.
"I am thankful to every player, every person who helped and was involved in the team. Many thanks to all the fans for their support and criticism.
"Together, we managed to show that our football can be competitive, productive and exciting. With faith in Ukraine."
Ukraine advanced as one of the best third-placed finishers in the group stage of Euro 2020 and beat Sweden 2-1 after extra time in the last 16.
However, their campaign was ended in the next round with a resounding 4-0 defeat to eventual runners-up England.
Russia's military crossed the border into neighbouring Ukraine on Thursday and commenced military action, prompting widespread condemnation.
Shevchenko, Ukraine's all-time leading goalscorer and former head coach, wanted people to join him in speaking out against the attacks as he expressed a desire for the restoration of peace.
"In the early hours, a full-scale war was initiated by Russia. My people and my family are under attack," Shevchenko wrote on social media.
"Ukraine and its population want peace and territorial integrity. Please, I ask you to support our country and call the Russian government to stop their aggression and violation of international law.
"We only want peace. War is not the answer."
